AN ABSTRACT OF THE DISSERTATION OF
Dannele E. Peck for the degree of
Doctor of Philosophy in
Agricultural and Resource Economics presented
on December 13, 2006.
Title: Economics of Drought Preparedness and Response in Irrigated Agriculture.
Abstract approved:
__________________________________________________________________
Richard M. Adams
The impact of recent severe droughts throughout the United States, the
potential for climate change to intensify the frequency and severity of drought, and
discussion about the future of government assistance in agriculture highlight the
need for a transition from drought as ‘disaster’ to drought as ‘managed risk’.
However, guidance for agricultural producers about optimal drought preparedness
and response is insufficient. It is particularly unclear what optimal drought
preparedness and response should look like, in practice, for farm systems with
uncertain water supplies and intra- and inter-year dynamics.
A mathematical programming model that captures the stochastic and
dynamic aspects of an irrigated row crop farm is developed and used to explore the
nature of optimal drought preparedness and response. Results indicate several
important characteristics. First, drought has the potential to generate
heterogeneous impacts, even across a set of homogeneous farms. Second, a farm
system with inter-year dynamics can continue to experience the effects of drought
after the drought itself subsides; additionally, the effects of drought in one year can
intensify the impact of drought in subsequent years. Third, in the presence of
discount and interest rates, crop diversification does not maximize expected profit,
even though it is often considered a drought management tool. Fourth, the primary
effect of water supply uncertainty is the abandonment of more fall-prepared fields.
Hence, the multi-peril crop insurance program’s prevented planting provision is
identified as an optimal drought preparedness tool, even if unsubsidized. Finally,
the predicted effects of climate change for snowmelt-dependent farm systems
require distinctly different forms of adaptation, and cause profit losses of different
magnitudes.
Because the model captures both intra- and inter-year dynamics, it provides
1) a more thorough understanding of the complex tradeoffs that producers face
when preparing for and responding to drought, 2) a more complete picture of the
dynamic impacts of drought, and 3) important insights about the administration of
drought assistance programs. Lastly, it elucidates the meaning of optimal drought
preparedness; a notion that has received increased attention in the policy arena, but
whose practical form has been only vaguely alluded to.
©Copyright by Dannele E. Peck
December 13, 2006
All Rights Reserved
Economics of Drought Preparedness and Response in Irrigated Agriculture
by
Dannele E. Peck
A DISSERTATION
submitted to
Oregon State University
in partial fulfillment of
the requirements for the
degree of
Doctor of Philosophy
Presented December 13, 2006
Commencement June 2007
Doctor of Philosophy dissertation of Dannele E. Peck
presented on December 13, 2006.
APPROVED:
Major Professor, representing Agricultural and Resource Economics
Head of the Department of Agricultural and Resource Economics
Dean of the Graduate School
I understand that my dissertation will become part of the permanent collection of
Oregon State University libraries. My signature below authorizes release of my
dissertation to any reader upon request.
Dannele E. Peck, Author
ACKNOWLEDGEMENTS
I thank my major professor, Richard Adams, for involving me in the
project that motivated this dissertation; for providing the means for me to interact
with agricultural producers in the study area; for encouraging independent
discovery, but providing timely guidance, and finally, for sharing his wisdom and
sense of humor.
Several other people contributed to this project, and I thank each of them
for so generously sharing their time, knowledge and data. I am particularly
grateful to Marshall English, Bill Jaeger, Greg Perry, and JunJie Wu for their
investment as committee members in this research project and in my intellectual
development. Their expertise, intuition, creativity, and professional success are
inspiring. I also thank Jeff Arthur for serving on my committee until leaving on
sabbatical. Several agricultural producers in the Vale Oregon Irrigation District
took valuable time from their hectic work days to show me the ins and outs of
managing an irrigated crop farm. That they contend daily with such complex
decision problems is humbling, and I hope that this dissertation contributes
practical insights about that decision environment. I thank the following people
who also helped me understand the farm systems in the study area: Scott Ward
and Barbara Kulhman with the Vale Oregon Irrigation District office; Janet
Haight, a crop insurance agent for several producers in the study area; Ron Jacobs,
the District 9 Watermaster for the Oregon Water Resource Department; Lynn
Jensen with the Malheur County Agricultural Extension Service; Clinton Shock
with the Malheur County Agricultural Experiment Station, and Scott Stanger, the
branch manager of Farm Credit Services in Ontario, Oregon. I am grateful to
Michael Bussieck, with GAMS Development Corporation, for helping me finetune the CPLEX solution algorithm, such that my largest model could be solved. I
also thank the USDA-Economic Research Service for providing the grant that
supported a portion of this research.
The friendship and camaraderie that my fellow graduate students provided
was invaluable. I feel very fortunate to have been part of such a considerate and
supportive group. I am particularly indebted to Branka Valcic, without whom I
would have learned much less about economics and life, and had significantly less
fun doing it. Thank you for laughing and crying through this experience with me.
Finally, I thank Ben Rashford for reminding me of my strengths when I had
forgotten them, for discounting my weaknesses when I felt most vulnerable to
them, for embarking on this adventure with me many years ago, and for continuing
to make me laugh after all these years. He deserves much of the credit for my
accomplishments.
TABLE OF CONTENTS
Page
1. INTRODUCTION ............................................................................................ 1
2. REVIEW OF RELEVANT CONCEPTS AND LITERATURE ...................... 8
2.1
DECISION-MAKING UNDER UNCERTAINTY .............................. 8
2.2
INCORPORATING STOCHASTICITY IN LINEAR
PROGRAMMING MODELS............................................................. 15
2.2.1 Introduction to Linear Programming ...................................... 15
2.2.2 Chance Constrained Programming ......................................... 17
2.2.3 Passive Programming.............................................................. 23
2.2.4 Stochastic Programming ......................................................... 25
2.2.5 Stochastic Dynamic Programming.......................................... 30
2.3
ECONOMIC STUDIES OF AGRICULRUAL
WATER SHORTAGE ........................................................................ 33
2.4
THE MULI-PERIL CROP INSURANCE PROGRAM’S
PREVENTED PLANTING PROVISION .......................................... 35
2.5
CLIMATE CHANGE AND DROUGHT IN THE WESTERN
UNITED STATES .............................................................................. 40
3. DESCRIPTION OF THE STUDY AREA...................................................... 43
3.1
OVERVIEW ....................................................................................... 43
3.2
RESEVOIRS ....................................................................................... 43
3.3
THE DISTRIBUTION OF WATER................................................... 49
3.4
AGRICULTURAL PRODUCTION................................................... 50
3.5
IRRIGATION TECHNOLOGY ......................................................... 51
3.6
WATER SUPPLY FORECASTS ....................................................... 52
3.7
SUMMARY ........................................................................................ 53
4. MODEL DESCRIPTION................................................................................ 56
4.1
OVERVIEW OF THE MODEL ......................................................... 56
4.1.1 The Binary Variables Model................................................... 58
4.1.2 The Continuous Variables Model ........................................... 60
4.2
INTERPRETING THE GENERAL MODEL .................................... 61
4.3
DETAILS OF THE BINARY VARIABLES MODEL ...................... 64
4.4
DETAILS OF THE CONTINUOUS VARIABLES MODEL............ 75
4.5
ALTERNATIVE VERSIONS OF THE BINARY VARIABLES
MODEL .............................................................................................. 82
4.5.1 Water Supply Certainty........................................................... 83
4.5.2 Price Uncertainty..................................................................... 84
TABLE OF CONTENTS (Continued)
Page
4.6
4.5.2.1
The General Price Uncertainty Model .................. 85
4.5.2.2
Details of the Price Uncertainty Model................. 90
4.5.3 Prevented Planting Provisions................................................. 96
MODEL VALIDATION..................................................................... 97
5. RESULTS AND DISCUSSION I: THE BASE CASE SOLUTION ........... 100
5.1
THE BASE CASE SOLUTION (Binary Variables Model) ............. 100
5.1.1 Optimal Cropping Activities ................................................. 102
5.1.1.1
Irrigation Technology Efficiency........................ 107
5.1.1.2
Open Fields in the Fall ........................................ 110
5.1.1.3
Fallowing ............................................................ 111
5.1.1.4
Deficit Irrigation ................................................. 112
5.1.2 Profit Outcomes .................................................................... 115
5.2
THE BASE CASE SOLUTION (Continuous Variables Model)...... 122
6. RESULTS AND DISCUSSION II: APPLICATIONS OF THE
BINARY MODEL ........................................................................................ 129
6.1
THE ROLE OF INTEREST AND DISCOUNT RATES ................. 129
6.2
THE ROLE OF UNCERTAINTY .................................................... 136
6.3
THE ROLE OF INTER-YEAR DYNAMICS .................................. 144
6.3.1 Single-Year Drought ............................................................. 145
6.3.2 Multi-Year Drought .............................................................. 151
6.4
THE ROLE OF CROP HISTORY ................................................... 155
6.5
PRICE UNCERTAINTY .................................................................. 161
6.6
PREVENTED PLANTING PROVISION OF THE MULTI-PERIL
CROP INSURANCE PROGRAM.................................................... 168
6.7
CLIMATE CHANGE ....................................................................... 181
7. SUMMARY OF RESULTS AND POLICY IMPLICATIONS ................... 191
BIBLIOGRAPHY ............................................................................................... 199
APPENDICES
............................................................................................... 214
LIST OF FIGURES
Figure
Page
2.1
Common preference scaling functions......................................................... 12
2.2
Preference scaling function or Bernoullian utility function, u(c),
for a risk-averse person ................................................................................ 12
2.3
Preference scaling function, u(c), for a risk-neutral person ......................... 13
2.4
Preference scaling function, u(c), for a risk-loving person.......................... 14
3.1
The Malheur River Basin, including Bully Creek Reservoir, Agency
(Beulah) Reservoir, and Warmsprings Reservoir, which serve the Vale
Oregon Irrigation District............................................................................. 44
3.2
Vale Oregon Irrigation District and neighboring irrigation districts,
Malheur County, Oregon ............................................................................. 45
4.1
Decision tree representation of decision-making under uncertainty............ 62
4.2
Decision tree representation of decision-making under water and
price uncertainty........................................................................................... 89
5.1
Optimal fall and spring activities in year 1 of the base case...................... 102
5.2
Optimal fall and spring activities in year 2 of the base case...................... 103
5.3
Optimal fall and spring activities in year 3 of the base case...................... 104
LIST OF FIGURES (Continued)
Figure
Page
5.4
Optimal fall and spring activities when the producer knows that the
next six years’ water allotments will be full versus dry............................. 106
5.5
Cumulative distribution function of the stream of discounted profit
for the base case solution ........................................................................... 116
5.6
Average change in total discounted profit (black dashes) by years
of drought experienced............................................................................... 118
5.7
Optimal fall and spring activities in year 1 of the continuous variables
model’s base case solution ......................................................................... 123
5.8
Optimal fall and spring activities in year 2 of the continuous variables
model’s base case solution ......................................................................... 124
5.9
Optimal fall and spring activities in year 3 of the continuous variables
model’s base case solution ......................................................................... 125
5.10 Average change in the continuous variables model’s total discounted
profit (black dashes) by years of drought experienced .............................. 126
6.1
Optimal fall and spring activities in year 1 of the “zero interest and
discount rates” case .................................................................................... 131
6.2
Optimal fall and spring activities in year 2 of the “zero interest and
discount rates” case .................................................................................... 132
6.3
Optimal fall and spring activities in year 3 of the “zero interest and
discount rates” case .................................................................................... 133
LIST OF FIGURES (Continued)
Figure
Page
6.4
(a) Average change in the “zero interest and discount rates” model’s
total discounted profit by years of drought experienced;
(b) A comparison of average change in total discounted profit for the
“base case” versus “zero rates case” .......................................................... 135
6.5
Cropping impacts of water supply certainty for scenario
[Full Full Full Full Full Full] ..................................................................... 139
6.6
Cropping impacts of water supply certainty for scenario
[Dry Dry Dry Dry Dry Dry]....................................................................... 142
6.7
Cropping impacts of a year 2 drought........................................................ 147
6.8
Crops assigned to each field in each year of the six-year planning
horizon ....................................................................................................... 149
6.9
Cropping impacts of a year 3 drought when preceded by a full versus
dry year 2 ................................................................................................... 154
6.10 Optimal fall and spring activities in year 1 of the “history” case .............. 157
6.11 Optimal fall and spring activities in year 2 of the “history” case .............. 158
6.12 Optimal fall and spring activities in year 3 of the “history” case .............. 159
6.13 Average change in the “history” model’s total discounted profit by
years of drought experienced ..................................................................... 161
LIST OF FIGURES (Continued)
Figure
Page
6.14 Optimal cropping activities for years 1 through 3 of scenario
[Full Full Full], as generated by a six-year versus three-year version
of the base case model................................................................................ 164
6.15 Optimal cropping activities for years 1 through 3 of scenarios
[Full Full Full] (price certainty) and [Full Med Full Med Full]
(price uncertainty), as generated by a three-year version of the
base case model.......................................................................................... 165
6.16 Cumulative distribution function of discounted profit without
prevented planting coverage and with subsidized PP coverage................. 177
6.17 Average change in total discounted profit by years of drought
experienced with and without subsidized PP coverage.............................. 178
6.18 Cumulative distribution function of discounted profit without
prevented planting coverage, with unsubsidized PP coverage, and
with subsidized PP coverage...................................................................... 180
6.19 Average change in total discounted profit by years of drought
experienced with subsidized, unsubsidized, and no PP coverage.............. 181
6.20 Cropping impacts of increased drought frequency on scenario
[Full Dry Dry Full Full Full]...................................................................... 184
6.21 Cropping impacts of increased drought severity on scenario
[Full Dry Dry Full Full Full]...................................................................... 185
LIST OF TABLES
Table
Page
2.1
A passive programming analysis of a hypothetical farm decision............... 24
2.2
Crop insurance programs available, and eligible crops, in the study area ... 39
3.1
Count and percent occurrence of five reservoir storage level classes
during water years 1969-2004 at Bully Creek, Beulah, and Warmsprings
Reservoirs..................................................................................................... 48
3.2
VOID water allotments (acre-feet per acre) for the years 1981-2003 ......... 50
3.3 Harvested acres by crop as a percent of total harvested acreage ................. 54
5.1
Parameter values assumed for the base case .............................................. 101
5.2
Summary statistics of the base case solution’ profit outcome ................... 116
5.3
Characteristics of change in total discounted profit ($) as the years
of drought experienced increases ............................................................... 119
5.4
Summary statistics of the base case’s profit outcome for the continuous
variables model .......................................................................................... 125
6.1
Number of fields successfully planted to various crops in scenario
[Full Full Full Full Full Full] under certainty and uncertainty .................. 138
6.2
Excess water by year for scenario [Full Full Full Full Full Full]
under uncertainty and certainty.................................................................. 138
LIST OF TABLES (Continued)
Table
Page
6.3
Number of fields attempted and successfully planted to various
crops in scenario [Dry Dry Dry Dry Dry Dry] under certainty and
uncertainty.................................................................................................. 141
6.4
Excess water by year for scenario [Dry Dry Dry Dry Dry Dry]
under uncertainty and certainty.................................................................. 143
6.5 Impact of a year 2 drought on undiscounted profit .................................... 148
6.6 Impact of a year 3 drought on undiscounted profit .................................... 148
6.7
Impact of a two-year drought on undiscounted profit................................ 153
6.8
Impact on undiscounted profit of a year 3 drought when preceded
by a year 2 drought..................................................................................... 153
6.9
Crop history used in the “history” model................................................... 156
6.10 Summary statistics of the “history” case’s profit outcome ........................ 160
6.11 Prices received by growers for yellow onions over the period
1995-2004 .................................................................................................. 162
6.12 Summary statistics of the profit outcome ($) for the following
three-year models: (i) water supply uncertainty and onion price
certainty, and (ii) water supply and onion price uncertainty...................... 168
LIST OF TABLES (Continued)
Table
Page
6.13 Unsubsidized and subsidized premiums, prevented planting
payment, and expected indemnity ($ per acre) for alternative
combinations of crop and MPCI coverage level ........................................ 171
6.14 Actuarial fairness of unsubsidized and subsidized multi-peril crop
insurance (MPCI) premiums, in 2004, for alternative crops and
coverage levels ........................................................................................... 173
6.15 MPCI coverage levels required to fully insure a producer’s losses
from drought-induced abandonment .......................................................... 174
6.16 Profit impacts of three climate change cases as compared to the
base case: 1) increase in the frequency of drought from 40 to 50%,
2) increase in drought severity from a 24 acre-inch/acre water
allotment to an 18 acre-inch water allotment, and 3) an increase in
drought frequency and severity.................................................................. 188
6.17 Average percent change in total discounted profit, by years of
drought experienced, for the base case and climate change
cases 1 through 3........................................................................................ 189
LIST OF APPENDICES
Appendix
Page
A
BASE CASE PARAMETERS AND PROFIT .......................................... 215
B
DESCRIPTION OF THE GAUSSIAN QUADRATURE PROCEDURE 237
B.1 DISTRIBUTION KNOWN .............................................................. 237
B.2 DISTRIBUTION UNKNOWN......................................................... 239
B.3 DISCRETE CATEGORIES FOR THE ONION PRICE .................. 240
C
PREVENTED PLANTING MODEL’S PARAMETERS AND PROFIT. 242
D
PRICE UNCERTAINTY MODEL’S PROFIT ......................................... 247
E
LIST OF ELECTRONIC FILES PROVIDED .......................................... 254
LIST OF APPENDIX FIGURES
Figure
Page
B.1 Beta distribution’s probability distribution function, overlaid
on a relative frequency histogram of the data ............................................ 238
B.2 Cumulative distribution function for the random variable
“water supply”, estimated with historical water allotment data
from the study area..................................................................................... 239
B.3 Cumulative distribution function for the random variable
“onion price”, estimated with historical data from the study area ............. 241
LIST OF APPENDIX TABLES
Table
Page
A.1 Per-unit prices and maximum yield assumed in the base case................... 215
A.2 Cost of crop production under alternative irrigation technologies............. 216
A.3 Crop water requirements ............................................................................ 218
A.4 Technical efficiency of alternative irrigation technologies........................ 219
A.5 Assumed yield response factor (ky) for alternative crops .......................... 220
A.6 Crop yield, total revenue, total cost , and net revenue for
alternative combinations of crop, irrigation technology, and
deficit irrigation level................................................................................. 221
A.7 Discounted and undiscounted profit for each water supply scenario
in the binary base case solution.................................................................. 227
A.8 Irrigation water requirement and net revenue per acre-inch of water
requirement for alternative crop-irrigation-deficit combinations .............. 230
C.1 Approved yields used in the calculation of a prevented
planting payment........................................................................................ 242
C.2 Available coverage levels for the multi-peril crop insurance contract ...... 242
C.3 Price election assumed for each insured crop ............................................ 242
LIST OF APPENDIX TABLES (Continued)
Table
Page
C.4 Prevented planting coverage level assumed for each insured crop............ 242
C.5 Discounted and undiscounted profit by scenario for the “subsidized
prevented planting” model’s solution ........................................................ 243
C.6 Discounted and undiscounted profit by scenario for the “unsubsidized
prevented planting” model’s solution ........................................................ 245
D.1 Discounted and undiscounted profit by scenario for the
“price uncertainty” model’s solution ......................................................... 247
Economics of Drought Preparedness and Response in Irrigated Agriculture
1
Introduction
Climate variability is a major source of uncertainty for agriculture in the
United States, generating an expected annual loss of $80-95 billion (Easterling and
Mendelsohn 2000). Drought is one manifestation of climate variability that
continues to challenge agriculture, particularly in the semi-arid regions of the
western United States, where the frequency of drought is high (Wilhite and Rhodes
1993). Tannehill (1947, p. 15) wrote of the unique nature of drought, particularly
the challenge of recognizing it in advance:
We have no good definition of drought. We may say truthfully that
we scarcely know a drought when we see one. We welcome the
first clear day after a rainy spell. Rainless days continue for a time
and we are pleased to have a long spell of such fine weather. It
keeps on and we are a little worried. A few days more and we are
really in trouble. The first rainless day in a spell of fine weather
contributes as much to a drought as the last, but no one knows how
serious it will be until the last dry day is gone and the rains have
come again. We are not sure about it until the crops have withered
and died.
Agricultural producers throughout much of the United States have recently
experienced the creeping nature of drought to which Tannehill eluded over 50
years ago. The 1999-2006 drought is one of the most severe in the last 100 years
(Heim and Lawrimore 2006). At the peak of the drought, in 2004, two-thirds of
the western United States was affected (Heim and Lawrimore 2006). Indices such
as the Standardized Precipitation Index (SPI), Palmer Drought Severity Index
(PDSI), Surface Water Supply Index (SWSI), and Crop Moisture Index (CMI)
have sharpened the definition and detection of agricultural drought, but the science
of drought prediction is still in its infancy (Dole 2000). Agricultural producers, as
a result, rely largely on past experience and providence to account for the risk of
2
drought in their farm management plans (i.e. to prepare for drought), and to adjust
farm plans in the event of drought (i.e. to respond to drought).
Research suggests that global climate change, particularly increased
evaporation rates, and a larger proportion of precipitation in the form of rain
versus snow, will enhance the frequency and intensity of drought in many areas of
the western United States (Gleick 2000; Intergovernmental Panel on Climate
Change 2001a; Intergovernmental Panel on Climate Change 1998). It will become
increasingly important, in the event of such climate changes, to understand the
characteristics of drought preparedness and response in farm management plans.
Drought preparedness and response has concomitantly appeared in the policy
arena, as policymakers discuss the future of government assistance in agriculture,
including, for example, subsidized crop insurance (a drought preparedness tool)
and disaster assistance programs (a drought response tool) (Knutson 2001;
Western Drought Coordination Council 1999). Australia set an extreme example
for U.S. policymakers in the late 1980s by removing drought from the list of
recognized natural disasters. Agricultural producers in Australia continue to
struggle with the policy-transition from drought as a ‘disaster’ to which they
simply respond, to drought as a ‘managed risk’ for which they prepare (Stehlik
2005). Although drought preparedness has garnered increased attention in the
U.S., guidance for producers about how to incorporate drought preparedness into
the farm’s broader management plan remains insufficient.
Economic studies have increased the understanding of drought
preparedness and response. However, simplifying assumptions about farm
systems’ uncertainty and dynamics, which are commonly used to improve model
tractability, result in an incomplete understanding of the many tradeoffs producers
face. Four alternative sets of assumptions are common: 1) certainty with no
dynamics (Adams and Cho 1998; Bernardo et al. 1987; Jaeger 2004; Michelsen
and Young 1993), 2) certainty with intra- or inter-year dynamics (Garrido and
Gomez-Ramos 2000; Haouari and Azaiez 2001; Iglesias, Garrido, and Gomez-
3
Ramos 2003; Thompson and Powell 1998), 3) uncertainty with intra-year
dynamics (Adams et al. 1995; Keplinger et al. 1998; Mejias, Varela-Ortega, and
Flichman 2004; Taylor and Young 1995; Turner and Perry 1997), and 4)
uncertainty with inter-year dynamics (and in some cases intra-year dynamics as
well) (Toft and O'Hanlon 1979; Weisensel, Van Kooten, and Schoney 1991). The
fourth set of assumptions most thoroughly captures the decision-making
environment of a producer who faces uncertainty and a dynamic farm system.
Few studies use this set of assumptions, however, to address optimal drought
preparedness and response (where “optimal” refers throughout this dissertation to
the solution that maximizes a mathematical programming model’s farm-level
objective function; nowhere in this dissertation is “optimal” used to indicate Pareto
optimality or social efficiency). No studies have used this set of assumptions in
the context of an irrigated crop farm.
Uncertainty and dynamics make it challenging for a producer to identify
optimal drought preparedness and response plans. Because of uncertainty,
producers typically do not know, prior to decision making, whether drought will
occur in the near future, when or how frequently it will occur in the more distant
future, how severe drought will be, or for how long any one drought will persist.
Because of intra- and inter-year dynamics, producers also have to consider how
current decisions will affect opportunities and outcomes in future periods.
Producers whose farm systems involve both uncertainty and dynamics have to
keep two things in mind when making drought preparedness and response plans:
1) the cost of drought preparedness includes foregone opportunities if drought does
not materialize, and 2) the dynamic effect of their plan on future decisions is statedependent. The challenge, in summary, is to determine “whether the long-run
rewards will be greater if one hedges against drought in their year-to-year
operations, or plunges ahead boldly, facing up to drought only when it actually
hits” (Clawson et al. 1980, p. 45). Given the considerable complexity of such a
4
decision environment, it is difficult to derive an optimal drought preparedness and
response plan based on intuition alone.
This dissertation helps to develop the intuition of optimal drought
preparedness and response by creating and solving a multi-year dynamic and
stochastic farm decision model to obtain an optimal plan and examining the
tradeoffs and parameters that shape that plan. The model is parameterized for a
hypothetical irrigated row-crop farm in the Vale Oregon Irrigation District (VOID)
of eastern Oregon. A row-crop system was chosen because it involves both intraand inter-year dynamics. The primary intra-year dynamic is that fall fieldpreparation and planting decisions affect spring planting decisions. The primary
inter-year dynamic is that the crop choice for a particular field in year t limits crop
choices for that field in future years, via agronomic constraints. VOID was chosen
as the study area for the following reasons: 1) water supplies for the upcoming
growing season are uncertain at the time fall decisions are made; 2) VOID
producers experience drought frequently (most recently a three-year drought that
ended in 2004), and have consequently adopted several preparedness and response
tools, and 3) producers have indicated a desire to enhance their ability to prepare
for and respond to drought.
The farm decision model can be modified in numerous ways to address a
suite of research questions. This dissertation focuses on research questions that
relate to three bodies of literature: farm-level drought preparedness and response;
drought-related farm policy, and mathematical modeling of stochastic and
dynamics farm systems. Gaps in these bodies of literature are identified in the
literature review. Research objectives include the following: 1) to understand the
tradeoffs that drive the optimal form and degree of drought preparedness and
response identified by the model, and generalize them for potential application in
other farm systems, 2) to determine the role of inter-year dynamics in the
management and impact of drought, 3) to explore the usefulness of the multi-peril
crop insurance program’s prevented planting provision, at the farm-level, as a
5
drought preparedness tool, and 4) to highlight the advantages and disadvantages of
using integer stochastic programming to model a stochastic and dynamic farm
system. An overview of research findings is provided next.
The optimal (i.e. expected profit-maximizing) form and degree of drought
preparedness and response is considered first. Drought preparedness is defined in
this dissertation as the means by which an agricultural producer plans for drought
before they know specifically when it will occur. This is in contrast to drought
response, which is defined here as an action taken once a producer knows that a
drought will occur. Drought response tools are shown to be part of the optimal
farm plan, but drought preparedness tools are also prevalent. The magnitude of
profit loss attributable to drought under optimal preparedness and response is
difficult to generalize because it exhibits large variation depending on the crops
planted at the time the drought occurs. Economic parameters, such as the interest
and discount rates, play an important role in the solution’s characteristics. The
effects of uncertainty are then examined to determine how the impacts of drought
differ when anticipated versus not. The primary effect of drought under water
supply uncertainty, in contrast to certainty, is identified, and shown to have
implications for the multi-peril crop insurance program’s prevented planting
provisions.
Inter-year dynamics is an important characteristic of many farm systems.
Yet the role of inter-year dynamics in drought preparedness and response, or its
implications for profit impacts of drought, is not well-understood. Producers
indicate that inter-year dynamics sometimes result in the persistence of drought’s
effects well after the drought subsides. The model’s solution is examined for
evidence of such persistence. Drought, and the response to it, affect cropping
plans and profit in subsequent years via inter-year crop dynamics. This result has
implications for the effectiveness of government assistance in response to drought.
It also has implications for the impact of multi-year droughts. The above results
provide a more complete understanding of the complex tradeoffs that producers in
6
a stochastic and dynamic farm system face when preparing and responding to
drought.
The ability of the multi-peril crop insurance program’s prevented planting
provision to mitigate the farm-level impacts of drought is also explored. The
prevented planting provision, which covers losses attributable to an anticipated
water shortage, is shown to be a cost-effective drought preparedness tool for
producers, whether premiums are subsidized or not. Enrollment in the prevented
planting provision effectively eliminates profit loss attributable to drought. No
attempt is made, however, to determine the social efficiency or social-costeffectiveness of the prevented planting provision.
The effect of climate change, specifically more frequent and severe
drought, on optimal drought preparedness and response, and on profit loss
associated with drought is also analyzed. An increase in drought frequency has
little impact on the drought preparedness plan or on profit loss attributable to
drought. This result does not hold, however, for an increase in drought severity (or
both severity and frequency). Adjustments in the crop plan, and change in profit
loss differ substantially depending upon which climate parameter changes. These
insights inform discussions about the need for and the design of government
assistance in a changing climate.
The last body of literature to which this dissertation contributes is
mathematical modeling of stochastic and dynamic farm systems. The use of multistage discrete sequential stochastic programming (DSSP) to capture the dynamic
and stochastic features of a farm system is illustrated. Few studies have taken
advantage of multi-stage DSSP’s ability to represent both intra- and inter-year
dynamics. A second contribution is made by solving both a binary and continuous
variables version of the model and comparing their solutions. A binary model
represents the producer’s decision problem more accurately than a continuous
model, but it is also more difficult to solve. The ability of a continuous model to
approximate the binary model’s solution is therefore examined. The producer has
7
more flexibility in the continuous model than in the binary model; therefore, the
continuous model fails to identify some of the drought preparedness tools
identified by the binary model.
The remainder of the dissertation is organized as follows. Chapter 2
provides an overview of relevant concepts and literature. Chapter 3 describes the
study area. Chapter 4 presents the farm decision model. Chapters 5 and 6 present
results and a discussion of farm-level implications. Chapter 7 summarizes
research findings and draws them together in a concluding discussion of potential
implications for drought-related farm programs.
8
2
Review of Relevant Concepts and Literature
This chapter discusses key concepts and literature relevant to the
exploration of the objectives of this dissertation. Specifically, chapter 2 consists of
the following sections: 2.1 Decision-making under uncertainty, 2.2 Incorporating
stochasticity in linear programming models, 2.3 Economic studies of agricultural
water shortage, 2.4 The multi-peril crop insurance program’s prevented planting
provision, and 2.5 Climate change and drought in the western United States.
2.1
Decision-Making under Uncertainty
Agricultural producers make many production decisions without knowing
the outcome a priori. Much of the uncertainty that agricultural producers face is
due to the strong influence of nature on the production of agricultural goods, and a
limited ability to predict nature. In addition to their subjective beliefs about the
probability of different states of nature, a producer’s physical and financial
resources, management objectives, and attitude towards risk also influence their
decisions under uncertainty.
Economists incorporate many of these factors in mathematical models to
improve their ability to mimic the complex process of decision-making under
uncertainty, and thus enhance the robustness of their economic analyses. Producer
characteristics are diverse, so economists typically must choose a finite number of
“representative” producers to model, but model performance varies with the
chosen characteristics. This fact is especially relevant for the selection of
management objectives and risk preferences. This section of the literature review
provides an overview of the standard approach to modeling decision-making under
uncertainty, and discusses the selection of management objectives and risk
preferences.
9
In the presence of uncertainty, a decision problem has the following
elements (Hirshleifer and Riley 1992, p. 7; Mas-Colell, Whinston, and Green
1995, p. 184):
1) a set of actions available to the decision-maker, (x = 1,…,X),
2) a set of states possible in nature, (s = 1,…,S),
3) a consequence function, c(x,s), showing outcomes of all combinations of
acts and states,
4) a probability function, p(s), expressing the decision-maker’s beliefs about
the likelihood of each state,
5) a preference scaling function, u(c), (also referred to as the Bernoullian
utility function), which measures the desirability of the different possible
consequences, and
6) a von-Neumann-Morgenstern expected utility function, U(x), which maps a
preference ordering for the set of acts from the preference scaling function
and probability function.
A researcher can usually identify elements 1), 2) and 3) with relative ease. The
researcher cannot, however, identify, a priori, how a decision-maker will choose
among the actions. This will depend on the remaining elements, 4), 5) and 6),
which can vary significantly between individuals, even when faced with the same
decision problem.
Since the true probability of the random states of nature (element 4) can
never be known with certainty, individual decision-makers must form subjective
probabilities. This subjective probability distribution must be elicited from the
decision-maker, or assumed by the researcher. The preference scaling function
(element 5) must also either be elicited from the decision-maker, or assumed by
the researcher. Once elements 4) and 5) have been established, element 6), U(x),
can be derived from them.
The decision-maker’s expected utility function, U(x), expresses their
preference for actions. Each action is associated with a set of potential
10
consequences; each consequence results from a different state of nature, and each
state of nature has a probability of occurrence. Under certainty, the decisionmaker knows precisely the consequence of each action, because the state of nature
has been revealed prior to the decision choice. In contrast, under uncertainty, the
decision-maker does not know which state will occur, and therefore does not know
for certain which consequence will be realized for each action. When choosing an
action given uncertainty, the decision-maker cannot simply choose the action that
is directly associated with their preferred consequence, because actions are no
longer necessarily associated with a single consequence.
The “Expected Utility Rule” of von Neumann and Morgenstern provides a
way to order preferences over actions when actions and consequences are not oneto-one (Hirshleifer and Riley 1992, p. 14). The Expected Utility Rule states that if
utility has the expected utility form (i.e. rational preferences satisfying the
continuity and independence axioms (Mas-Colell, Whinston, and Green 1995, p.
175)), and we are given the following information,
1) uncertain states of nature (1,…,s,…S),
2) consequences of action x under each state (cxs),
3) the probability of each state (ps), and
4) the utility of each consequence (u(cxs)),
then the utility gained from action x is U(x), where:
S
U(x)= U(c x1 , c x2 ,...,c xS ; p1 , p2 ,...pS )= p1u(c x1 )+p 2 u(c x2 )+...+pS u(c xS ) = ∑ ps u ( cxs ) ,
s=1
(Hirshleifer and Riley 1992, p. 14). Simply, the utility from action x is equal to
the probability-weighted average (i.e. mathematical expectation) of the utilities of
the consequences associated with action x, for all states of nature. In terms of the
selection of a preferred action, the “Expected Utility Theorem” states that for
preferences that admit the expected utility form, action x is strictly preferred to
action y if:
11
S
U ( x ) = ∑ ps u ( cxs ) >
s =1
S
∑ p u(c
s =1
s
ys
) = U ( y)
(Mas-Colell, Whinston, and Green 1995, p. 176). The Expected Utility Rule
translates the preference scaling function and the probability of states into a vonNeumann-Morgenstern expected utility function. The Expected Utility Theorem
provides a rule for choosing among actions.
Economists are interested in the decision-maker’s utility function, since it
contains information about behavior that can be used to predict their choice of an
action. Differential calculus is often used to describe the utility function, but since
actions are often discrete, differential calculus cannot be used to describe the
properties of U(x). Consequences, however, (usually expressed as monetary
values) are often continuous; hence, economists look at the properties of the
preference scaling function (Bernoullian utility function), u(c), to deduce the
decision-maker’s behavioral properties. From this point forward, the preference
scaling function, u(c), is used to define and discuss behavioral properties, such as
risk-attitude. Since U(x) is a linear combination of points on u(c), it is appropriate
to use u(c) for the purpose of determining behavioral properties.
The shape of an individual’s preference scaling function, and hence the
utility function, is as unique as the individuals themselves. Figure 2.1 shows three
shapes commonly assumed for the preference scaling function. The preference
scaling function is usually assumed to be upward sloping; this implies monotonic
utility, i.e. as the quantity of a desirable consequence increases, utility increases, or
u '(c) =
δ u (c )
> 0.
δc
The shape of the preference scaling function indicates the decision-maker’s
risk attitude. Figure 2.2 reveals that person A, represented by curve uA, when
faced with the choice between a gamble, G(c1= -$200, c3= +$200; p1=0.5, p3=0.5),
with an expected consequence of c2=$0 (a fair gamble), or a guaranteed
consequence of c2=$0, derives greater utility from the guaranteed consequence
12
uA
u(cxs)
100
u
B
u
C
40
5
0
c xs
c =$100
Figure 2.1. Common preference scaling functions (Hirshleifer and Riley
1992, p. 24).
uA
u(cxs)
uA($0)
uA(gamble)
- $200
$0
$200
xs
CE(gamble)
Figure 2.2. Preference scaling function or Bernoullian utility function,
u(c), for a risk-averse person.
than from the gamble, even though the gamble’s expected consequence is equal to
the guaranteed consequence. Person A is risk-averse. They would accept a
guaranteed amount that is less than the gamble’s expected payout (their certainty
equivalent, CE), rather than take the gamble.
13
Figure 2.3 reveals that person B, represented by curve uB, when faced with
the choice between a gamble with an expected consequence of c2=$0 (a fair
gamble), and a guaranteed consequence of c2=$0, derives equal utility from the
two choices. Person B is risk-neutral. They only care about the expected payout of
a gamble, and do not care about the variance of the payout. Figure 2.4 reveals that
person C, represented by curve uC, when faced with the choice between a gamble
with an expected consequence of c2=$0 (a fair gamble), and a guaranteed
consequence of c2=$0, derives less utility from the guaranteed consequence than
from the gamble. Person C is risk-loving. They would have to be paid a
guaranteed amount that exceeds the gamble’s expected payout to not take the
gamble; this amount is their certainty equivalent of the gamble, or CE(gamble).
uB
u(cxs)
uB($0) = uB(gamble)
- $200
$0
$200
cxs
Figure 2.3. Preference scaling function, u(c), for a risk-neutral person.
Risk-preferences are expressed as the rate of change of the slope of the
preference scaling function, equivalently, the second derivative of u(c),
u ''(c) =
∂ 2 u (c )
. Strict risk aversion is expressed as the strict concavity of u(c), i.e.
∂c 2
u '(c ) > 0, u ''(c ) < 0 . Strict concavity implies that the person’s marginal utility of
14
money is decreasing. That is, at any level of wealth, the utility gained from having
an additional dollar is less than the utility lost from having one fewer dollar (MasColell, Whinston, and Green 1995, p. 186). In a gamble where equal amounts of
money can be lost
u(cxs)
uC
uC(gamble)
uC($0)
=
- $200
$0
$200
cxs
CE(gamble)
Figure 2.4. Preference scaling function, u(c), for a risk-loving person.
or won, the disutility of losing outweighs the utility of winning, so the gamble is
not taken. Risk neutrality is expressed as a linear u(c), i.e. u '(c) > 0, u ''(c) = 0 . A
risk-neutral person has a constant marginal utility of money. That is, their utility
from an additional dollar is equivalent to their disutility from losing one dollar. In
a gamble where equal amounts of money can be lost or won, they are indifferent
between taking or not taking the gamble. Strict risk loving is expressed as the
strict convexity of u(c), i.e. u '(c) > 0, u ''(c) > 0 . The risk-lover’s marginal utility
of money is increasing. That is, the utility gained from having an additional dollar
is more than the disutility of losing a dollar; thus, the risk-lover chooses to take the
gamble when equal amounts of money can be lost or won.
15
It is clear that the preference scaling function’s general shape, or functional
form, has important implications for the decision-maker’s behavior under
uncertainty. A review of the literature on decision-making under uncertainty
reveals that agricultural producers are commonly assumed to be, or found through
empirical analysis to be, either risk-averse or risk-neutral (Gomez-Limon, Arriaza,
and Riesgo 2003; Hardaker, Huirne, and Anderson 1997, p. 101; Lin, Dean, and
Moore 1974; Meyer 2002; Torkamani and Haji-Rahimi 2001). The choice
between risk-averse or risk-neutral depends on the decision-making scenario, but
has important implications for the structure of the decision-making objective.
Specifically, when profit is the only argument in the utility function, risk-neutrality
implies that the producer’s objective is to maximize expected profit. Riskaversion, in contrast, implies that the producer cares not only about expected
profit, but also about the variance of profit. An expected utility maximization
problem with a non-linear utility function would be developed for the risk-averse
producer.
The choice of whether to model a producer as maximizing expected utility
or expected profit can have significant implications for the model’s solution (Isik
2002; Just 1975). A non-linear utility function often increases the complexity of
analyses, particularly for mathematical programming models. This source of
complexity may require simplification of other, more critical, areas of the model.
Risk neutrality is assumed in this dissertation, so that the stochastic and dynamic
features of the model can be enhanced. Pannell et al. (2000) suggest that the
marginal benefit of accommodating risk aversion is small relative to that of
improving other aspects of a farm-level mathematical programming model.
2.2
2.2.1
Incorporating Stochasticity in Linear Programming Models
Introduction to Linear Programming
Linear programming (LP) is a common tool for analyzing farm
management problems. LP models can be easily constructed and manipulated to
16
simulate many management scenarios. Equations (1) – (3) represent a generic LP
model.
(1) Min z = cT x
x
s.t.
(2)
Ax ≤ b
(3)
x≥0
where x is a vector of decision (activity) variables, and c, A, and b are vectors and
matrices (lowercase and uppercase letters, respectively) containing known
constants. The elements of vector c often represent, in a farm management
problem, the per-unit cost of activities in vector x. Matrix A contains technical
coefficients that express, for example, resource use per unit of x. Vector b would
then represent the quantity of resources available for use. Note that equation (2)
can be equivalently expressed in summation notation, as follows, for resource i
n
and activities j:
∑a x
ij
j
≤ bi i = 1, 2,..., M .
j=1
Assumptions underlying this general LP model limit the modeler’s ability
to realistically represent actual decision-making processes. The assumption of a
linear objective function, for example, precludes the use of a nonlinear utility
function to represent risk aversion. Nonlinear programming and linear
approximation approaches, such as MOTAD, have been developed to address this
limitation (Hardaker, Pandey, and Patten 1991). The assumption that all
coefficients and relationships are known is also unrealistic in some cases.
Mathematical programming models, including LP, are often unable to reproduce
producer behavior, in part because the model does not capture all elements of the
decision environment, or because the model is incorrectly parameterized.
Sensitivity analysis is often used in these cases to provide a range of possible
model outcomes. Alternatively, a calibration procedure known as positive
mathematical programming can be used to improve a model’s ability to replicate
reality (Howitt 1995).
17
The limiting assumption of interest in this study is deterministic
coefficients (c, A, and b in the equations above). Such an assumption essentially
ignores that farm managers make decisions in an uncertain environment. The
elements of c, A, and b often represent input and output prices, yield, and resource
availability and requirements, many of which are random variables whose values
are not revealed until after decisions are made. The assumption that all
coefficients are both constant and known a priori is unrealistic, and can generate
solutions that lead to suboptimal outcomes when applied in the presence of
uncertainty.
Chance-constrained programming, passive programming, and stochastic
programming are common approaches to relaxing the certainty assumption (i.e.
incorporating stochasticity) in farm management LP models. In the next section,
an overview of each approach is provided, followed by a discussion of their
advantages and weaknesses. The review concludes by identifying the approach
chosen for this study.
2.2.2
Chance-Constrained Programming
Chance-constrained programming (CCP) was an early attempt to introduce
stochasticity into mathematical programming problems (Charnes and Cooper
1959). Inspiration for the approach was derived from the oil industry’s need to
optimally schedule oil production subject to stochastic demand for heating oil
(Charnes, Cooper, and Symonds 1958). Industry, more specifically, sought a
production schedule that would guarantee that stochastic demands for heating oil
would be met with some level of probability.
CCP optimizes (i.e. maximizes or minimizes) the objective function
through choice of activity levels, subject to constraints, some of which involve
random variables with known distributions. The optimal activity levels must meet
all deterministic constraints, as well as maintain at a prescribed level of probability
18
the constraints that involve random variables (Charnes and Cooper 1959). The
following equations represent a generic CCP model.
(4)
Min z = cT x
x
s.t.
n
(5)
∑a x
ij
j
≤ bi
i = 1, 2, ..., m
j=1
n
(6)
P(∑ a hj x j ≤ b h ) ≥ p h
h = m+1, m+2,..., M
j=1
(7)
xj ≥ 0
where j represents activities, and i and h represent resources required for the
production of the activities. Subscript i represents resources that involve no
uncertainty. The quantity of resource i that is available for use and the quantity
required by an activity are constant and known a priori. Subscript h represents
resources that involve some uncertainty, perhaps in the quantity available, the
quantity required per unit of activity, or both. P is the probability operator, and ph
is some critical probability level pertinent to the constraint on the hth resource; the
decision-maker selects ph in advance.
Constraints that strictly contain deterministic coefficients are represented
by equation (5), just as they were in the LP model. Constraints containing either
random right-hand side or left-hand side coefficients, or both, are represented by
equation (6). Equation (6) states that the use of the hth resource across all j
activities must be less than the limit bh, with a probability of at least ph (Anderson,
Dillon, and Hardaker 1977 p. 222). Right-hand side stochasticity occurs when bh
is a random variable; that is, when the quantity of resource h available for use is
uncertain at the time decisions are made, but follows a known distribution. Total
water supply for the growing season is an example in farm management of a righthand side random variable. Its quantity varies annually and is rarely known at the
time planting decisions are made, but historical data can be used to estimate its
19
distribution. Dillon (1999), Maji and Heady (1978), and Keith et al. (1989)
provide additional examples of right-hand side stochasticity.
Left-hand side stochasticity occurs when ahj is a random variable; that is,
when the quantity of resource h required per unit of activity j is uncertain at the
time decisions are made, but follows a known distribution. Irrigation water
required per acre of crop j is an example of a left-hand side random variable. The
quantity of irrigation water required by a crop for the growing season varies with
factors that are highly unpredictable at the time planting decisions are made, such
as summer rainfall, air temperature, and humidity. Segarra, Kramer, and Taylor
(1985), Johnson and Segarra (1995), and Wojciechowski et al. (2000) provide
additional examples of left-hand side stochasticity. Most examples in the literature
consider the case of a single random variable, since having multiple random
variables in the same constraint may require their joint distribution.
The probabilistic structure of equation (6) must be converted to a
deterministic equivalent in order for a linear programming algorithm to find a
solution. Conversion involves the following steps: 1) estimate or assume the
random variable’s probability density function, 2) choose the desired critical
probability level, ph, at which the constraint should hold, and 3) determine the
value of the stochastic coefficient (ahj) or (bh), at which the constraint will hold ph
percent of the time, using the random variable’s density function. Assuming lefthand side stochasticity, the deterministic equivalent of equation (6) is represented
by the following equation:
(6a)
n
∑ aα x
hj
j
≤ bh
h = m+1, m+2,..., M
j=1
where, aαhj represents the value selected from ahj’s density function, such that the
constraint is guaranteed to hold ph percent of the time. Assuming right-hand side
stochasticity, the deterministic equivalent of equation (6) is represented by the
following equation:
20
(6b)
n
∑a
hj
x j ≤ bαh
h = m+1, m+2,..., M
j=1
where bαh represents the value selected from bh’s density function, such that the
constraint is guaranteed to hold ph percent of the time. Note that equations (6) and
(6a) are written such that the coefficient ahj is random for every activity j. This is
not always the case; a mixture of deterministic and random ahj is allowed. For
example, per acre yield of corn could be known and constant, while per acre yield
of wheat could be a random variable.
The intuition behind CCP is relatively straightforward. If a random
coefficient’s value is unknown at the time a decision must be made, and the
penalty of not meeting a constraint is severe, then you should assume a value for
the random variable such that the model’s solution, under most circumstances, will
meet the constraint. The previous example of an uncertain total water supply
(right-hand side) illustrates the intuition.
Suppose that the water supply’s expected value is 20 inches, but that it
varies from 10 to 40 inches, and follows a uniform distribution. If the random
variable “water supply” (bwater) is replaced with its expected value (this is known
as the “expected value problem” and denoted EV), the problem is equivalent to a
deterministic LP model. The solution indicates optimal activity levels given a
water supply of 20 inches. It does not indicate, however, the outcome of applying
these activity levels when water supply is revealed to be 10 inches, rather than 20.
The constraint that water use must not exceed water supply is likely violated, since
the optimal crop combination given 20 inches of water almost certainly consumes
more than 10 inches of water.
Suppose instead that the random variable bwater is replaced with the
conservative value of 10 inches (conservative in the sense that according to bwater ~
U[10,40] there is a high probability that in most years actual water supply will
exceed 10 inches). The resulting optimal activity levels, when implemented in the
presence of all possible realizations of water supply, would always result in the
21
water constraint being met. Essentially, by planning to be constrained to a small
quantity of a resource, the resulting optimal activity levels are such that the
resource constraint is met for most realizations of the random variable. Similarly,
in the case of a random left-hand side coefficient, by planning for a large quantity
of resource use per unit of activity, you choose activity levels that, under most
realizations of the random variable, result in the constraint being met (i.e. resource
use is less than or equal to resource supply).
How far the value chosen for the random variable deviates from its
expected value depends on the probability, ph, with which the modeler wishes the
constraint to be met. The closer ph is to 1, the more extreme the chosen value will
be. Calculating the appropriate value for a random variable given a pre-selected ph
involves estimating the mean and standard deviation, if the random variable is
distributed normally (see Segarra, Kramer, and Taylor 1985). If the distribution is
not normal, but relevant historical data are available, a value is chosen based on its
percentile (see Keith et al. 1989). For example, suppose a value representing the
25th percentile of historic water supplies, call it b25%, is chosen. Then by
definition, actual bwater will be less than b25% 25% of the time (and planned water
use will exceed actual water supply), and actual bwater will exceed b25% 75% of the
time (and planned water use will be less than actual water supply). Dillon (1999)
presents a slightly different approach for determining the appropriate value for the
random variable using historical data.
The simplicity of CCP is appealing. The most challenging steps are to
identify the random variable’s probability density function (or a set of historical
realizations), choose the desired level of ph, and select the appropriate
deterministic value for the random variable. Once these tasks are completed, the
problem is easily converted to its deterministic equivalent and solved with a
standard LP algorithm. Major criticisms of CCP exist, however. The first
criticism is that CCP deals only with random variables that appear in constraints,
and not those relevant to the objective function (Cocks 1968). It is not clear
22
whether this criticism is contradicted by the ability to construct a CCP model with
the following structure (Charnes and Cooper 1963).
(8)
Min P(cT x ≤ c0T x 0 )
x
s.t.
n
(9)
∑a x
ij
j
≤ bi
i = 1, 2, ..., m
j=1
n
(10) P(∑ a hj x j ≤ b h ) ≥ p h
h = m+1, m+2,..., M
j=1
(11) x j ≥ 0
The objective in this CCP problem is to choose activity levels that guarantee an
achieved value for the objective function with some probability, and meet resource
constraints with some probability (Cocks 1968).
A second criticism of CCP is the arbitrary selection of a value for ph. In
reality, the decision-maker selects a value for ph by weighing the tradeoff between
more certainty and less profit (Anderson, Dillon, and Hardaker 1977 p. 224;
Hardaker, Pandey, and Patten 1991). The selection of a value for ph should
therefore be part of the optimization problem. Askew (1974), in an attempt to
formulate a dynamic chance-constrained programming problem, may provide a
means for incorporating the choice of ph in the decision process. A final criticism
of CCP is that it assumes that constraint violation is acceptable (1-ph)% of the
time, but does not indicate what to do or to expect when constraints are actually
violated (Cocks 1968). Consider, for example, what might happen if a large city
managed its stochastic water supply in a manner that prevented shortages 9 out of
10 years, but failed to develop a management plan for the 1 year out of 10 when a
water shortage occurred. The CCP approach, while an improvement upon the
deterministic LP approach, clearly results in an incomplete solution to a stochastic
problem.
23
2.2.3
Passive Programming
It is assumed in the passive programming (PP) approach that optimization
will take place under certainty at a future date. The problem in the meantime is to
characterize the distribution of outcomes from which the eventual outcome will be
realized (Cocks 1968). The distribution of outcomes is derived by solving a
deterministic LP problem for each realization of the random variable. The general
notation for a PP problem follows.
(12) Min z(x,s) = csT x
x
s.t.
(13) A s x ≤ bs
(14) x ≥ 0
(15) ∀ s
The subscript, s, identifies coefficients whose values vary depending upon the state
of nature being considered. Suppose, for example, that the cost of planting an acre
of corn (an element of vector c) differs between two states of nature. The
appropriate value of the cost coefficient is c1 when state of nature 1 occurs, and c2
when state of nature 2 occurs. Similarly, coefficients in matrix A or vector b may
vary with the state of nature.
Equations (12) through (14) are solved for every state of nature, s. The
distributions of the optimal activity levels, x*(s), and objective values, z(x*(s),s),
are then formed from the collection of s solutions (Birge and Louveaux 1997 p.
138). A decision-maker could use these distributions to inform other management
decisions. The expected value of the objective function (known as the wait-andsee (WS) solution) is also of interest, and is calculated as follows (Cocks 1968):
(16) WS = Es z(x*(s),s) . See Tintner (1960) for a simple applied example, and
see Birge and Louveaux (1997 p. 138) for a more thorough treatment of equations
(12) through (16).
The PP approach represents a decision-maker who knows the random
variable’s realized value at the time the decision is made, and can therefore make
24
the decision based on perfect information. They do not know, however, the
variable’s value at the time they solve the PP problem. The best they can do, prior
to the decision-making period, is to solve the PP problem to determine the
distribution of the outcome, and use it to form an expectation.
The scenario described above is the appropriate situation in which to use
the PP approach. However, decision-makers might also use the PP approach to
inform their decisions when uncertainty is not resolved at the time decisions are
made. The distributions derived from PP may be combined with a heuristic rule to
guide decisions. Assume that a farm manager must make a decision before
uncertainty is resolved. They have conducted a PP analysis of their decision; the
solutions appear in Table 2.1. Suppose the manager constructs the following
decision rule: implement the activity level that occurs most frequently among all
the solutions. From the table of solutions, the manager would always plant 0 acres
of corn and 100 acres of wheat, because it appears most frequently among the set
of PP solutions. This rule obviously has no theoretical basis for being optimal,
which is the criticism of using the PP approach in scenarios where uncertainty is
not resolved a priori. It may, none-the-less, seem like a rational approach to a
busy manager.
Table 2.1. A passive programming analysis of a hypothetical farm
decision.
PP’s solution
Corn (ac)
Wheat (ac)
Very dry
0
100
State of Nature
Dry
Average
Wet
0
50
75
100
50
25
Very wet
100
0
PP is essentially the analysis of an LP problem’s sensitivity to different
values of the random variable (Higle and Wallace 2003). The approach’s
advantage is its simplicity. PP requires the modeler to solve one LP problem for
each possible value of the random variable. Once the initial LP model is
developed, the PP problem only involves repeatedly changing a single value in the
25
program and resolving it. Even if a random variable has hundreds of possible
values, PP requires only additional time, and not additional programming skill.
PP is not an ideal approach to incorporating stochasticity in LP, since its
assumption of perfect information at the time decisions are made is unrealistic.
Rae (1971) indicates, however, that the simple PP approach can be an
improvement over the expected value problem approach (EV) (defined in the
previous section). An additional role for PP exists, even in a world of imperfect
information. The perfect information solutions derived in PP provide a baseline to
which imperfect information solutions (such as those discussed next in the
stochastic programming section) can be compared. Such comparisons reveal the
expected value of perfect information (Birge and Louveaux 1997 p. 137), a topic
of interest to organizations who provide weather and price forecasts, for example.
2.2.4 Stochastic Programming
Stochastic programming (SP), also known as discrete stochastic
programming (DSP) and discrete sequential stochastic programming (DSSP) was
introduced by Cocks (1968) as a method for solving LP problems that include any
number of random variables as coefficients in the constraints and/or the objective
function. The ability to include random coefficients in constraints and the
objective function enables a modeler to account for the timing of decisions relative
to the timing of information discovery. Specifically, a modeler can represent a
multi-stage problem where decisions are made both before and after random
variables are realized. Decisions made before the random variables’ values are
revealed are known as “first-stage” activities, and denoted by a vector x.
Decisions or calculations made after random variables’ values are revealed are
known as “second-stage” or “recourse” activities, and denoted by a vector y. The
number of stages in a SP problem depends upon the number of
decision/information/recourse decision cycles that occur during the decisionmaking process. SP can also be adapted to single-stage problems where decisions
26
are made prior to random variables’ values being revealed, but no recourse
decisions are available (Cocks 1968).
An SP model consists of the following four pieces (Kaiser and Apland
1989): 1) a sequence of decision stages, 2) a set of decision variables for each
stage, 3) discrete or continuous random variables, and 4) an information structure
that represents the flow of information relative to the timing of decisions. SP
represents the following decision process. First-stage decisions are made prior to
random variables’ values being revealed. After decisions are made, the random
variables’ values are revealed. This resolution of uncertainty prompts secondstage decisions, with which the decision-maker attempts to fix sub-optimal
outcomes resulting from imperfect first-stage information. Rae (1971), in one of
the first applications of discrete stochastic programming to agriculture, illustrates
how SP captures this decision process.
The general goal of SP is to choose first-stage activity levels that minimize
current costs (or maximize current benefits) plus the expected cost (or expected
benefits) of second-stage activities. The SP solution indicates optimal first-stage
activity levels, as well as optimal second-stage activity levels for each possible
realization of the random variables. This approach, at least in its discrete form, is
reminiscent of decision tree analysis (Hardaker, Huirne, and Anderson 1997 p.
198). SP is unique among the approaches reviewed in this dissertation because it
suggests what to do after a specific state of nature is realized.
The SP model can be expressed in either the extensive form or the implicit
form, both of which are presented below. A single discrete random variable and a
two-stage stochastic program with recourse are assumed in the following examples
for notational ease. The extensive form SP model, shown below in equations (17)
through (20) (see also Birge and Louveaux 1997 p. 156), includes first-stage
decision variables, indexed by activity only (e.g. xi in the model below), and
second-stage decision variables, indexed by activity and state of nature (e.g. yis in
the model below). This form is labeled “extensive” because a set of second-stage
27
decision variables and constraints exists for every state of nature. Numerical
examples of the extensive form model can be found in Higle and Wallace (2003)
and Birge and Louveaux (1997 p. 8).
(17) Max z = -∑ ci x i +
x, y
i
∑∑ (r y
is
i
is
)ps
s
s.t.
(18)
∑x
i
≤L
(land constraint)
i
(19) yis ≤ a i x i
∀ s and i
(sales constraint)
(20) x i , yis ≥ 0
where,
x i = a first-stage decision variable for the i th activity
(e.g. acres of land devoted to crop i, where i = corn, wheat)
ci = the first-stage coefficient (known a priori ) associated with activity x i
(e.g. cost associated with devoting an acre of land to crop i)
yis = a second-stage decision variable for the i th activity when the s th state
of nature occurs (e.g. tons of crop i to sell when state s occurs)
s = state of nature of a random variable
(e.g. if output price is the random variable then s = low, average, high)
ris = the second-stage coefficient associated with activity yis
(e.g. price received per ton of crop i sold when the s th state
of nature occurs)
ps = the probability of the s th state of nature
L = limit on activity i (e.g. total acreage available)
a i = coefficient linking first-stage activity levels to second stage activity levels
(e.g. per acre yield of crop i)
The objective function above demonstrates SP’s goal of optimizing over
both the current (first-stage) costs and the expected value of the future (secondstage) revenues. The expectation is taken over the probability distribution of the
random variable. Note that while the collection of first-stage activities is [x1, x2],
the collection of second-stage activities is [y1 low, y1 avg, y1 high, y2 low, y2 avg, y2 high].
28
Second-stage activities, yis, are interpreted in this example as the tons of crop i to
sell when state of nature s occurs. As the notation above indicates, constraint (19)
is duplicated for every combination of crop i and state s. It is interpreted as
limiting the quantity of crop i sold when state s occurs to no more than the total
quantity yielded from acres planted to crop i in the first stage (xi). The set of
constraints represented by equation (19) establishes the timing of the decision
problem by linking first and second-stage activities. Constraints that apply only to
first-stage decision variables, such as equation (18), are not duplicated because
they do not vary across states of nature. The extensive form is limited, in practice,
to problems that involve few random variables and random variables with few
realizations. Otherwise, the “curse of dimensionality” arises, with symptoms
including cumbersome notation and myriad constraints (Anderson, Dillon, and
Hardaker 1977 p. 229; Hardaker, Huirne, and Anderson 1997 p. 197).
The implicit form of SP enables modelers to compress large stochastic
programming problems. The approach essentially tucks objective function terms
and constraints associated with second-stage variables into a sub-problem, denoted
Q(x,s), which represents the value of the second stage for a given realization of the
random vector s. In contrast to the previous assumption of a single random
variable, s, here a vector of random variables, s, is assumed. Boldface notation
indicates the vector is random, to differentiate it from its realization. The subproblem, Q(x,s), is optimized over second-stage decision variables for all
individual realizations of s and all feasible values of first-stage activities. The goal
of a single optimization of Q(x0,s0) is to select the optimal level of second-stage
activities, given the pre-selected first-stage activity level, x0, and the pre-selected
realization of s, s0. This optimization is repeated for all combinations of (x,s).
The expected objective value of all sub-problems, Q(x) = E Q(x,s), also known as
s
the value function or recourse function, is then placed into the SP problem’s
original objective function. The original SP problem is finally optimized over
29
first-stage decision variables, subject to first-stage constraints. The implicit form
is represented in the equations below.
(21) Min cT x + Es Q(x,s)
x
s.t.
(22) Ax = b
(23) x ≥ 0
where,
(24) Q(x,s) = Min qTy
s.t.
(25) Tx + Wy = h
(26) y ≥ 0,
where qT, hT, and T form the vector s, and contain the values taken by q, h, and T
under each state of nature, and where W is assumed here to be constant across
states of nature (fixed recourse). Birge and Louveaux (1997 p. 11) provide a
numerical example of the second-stage sub-problem that demonstrates the notation
used above.
The implicit form is less intuitive than the extensive form, but it reduces
the volume of programming code required, and is thus a more computationally
efficient approach to large SP problems (Birge and Louveaux 1997 p. 155). This
is especially useful when several random variables are involved, or when random
variables are continuous (see Birge and Louveaux 1997 p. 11). The extensive
form problem can be solved in the same manner as a generic LP problem. The
special structure of the implicit form, however, requires an alternative solution
algorithm. The L-shaped method is the most frequently used approach (see Birge
and Louveaux 1997 p. 156). It involves a three-step process that is repeated until
an optimal solution is found. The details of this approach are omitted here, given
the choice to focus on the more intuitive extensive form, rather than the implicit
form.
30
The most significant advantage to incorporating stochasticity into an LP
model via SP is that the approach captures the timing of decisions relative to the
flow of information more realistically than CCP and PP. Modelers using SP are
still limited to a relatively small number of stages, compared to the large number
of stages in a real farm manager’s decision-making process. However, Rae (1971)
demonstrates that the solution obtained from a three-stage SP model of a farm
manager’s decision process is expected to generate 16% more profit annually than
the solution obtained from an expected-value model (in which all random variables
are replaced with their expected values).
One disadvantage of SP is that it becomes more computationally difficult
as the number of random variables, realizations, and stages increase. SP also
requires the modeler to obtain more data to sufficiently represent the random
variables’ realizations and probability distributions. Finally, there is no general
rule for predicting the magnitude of gains from using the SP solution versus
solutions obtained from less sophisticated approaches (Birge and Louveaux 1997
p. 144). It is generally believed that stochastic programming is more relevant
when there is more randomness in the problem, but even this varies on a case-bycase basis. The benefit of modeling a decision-process using SP, rather than a
simpler approach, cannot be known a priori. The modeler might therefore invest a
great deal of time in developing an SP model only to discover that it produces a
solution that closely resembles those obtained from simpler approaches.
2.2.5
Stochastic Dynamic Programming
Stochastic dynamic programming (SDP) is similar to stochastic
programming (SP); however, the approaches have different strengths and
limitations. SDP, like SP, is a mathematical optimization technique for solving
multi-stage problems in which decisions are made under uncertainty. The general
form of an SDP problem is as follows (Kennedy 1986, p. 52):
31
⎡m
⎤
Vi { xi } = max ⎢ ∑ pi {ki } ai { xi , ui , ki } + aVi +1 {ti { xi , ui , ki }} ⎥ (i = n,...,1)
⎣ k =1
⎦
(
)
subject to
m
∑ p {k } = 1
i
i
k=1
with
Vn +1 { xn +1} = F { xn +1}
where,
Vi = objective or value function in decision stage i
xi = vector of states at stage i
(e.g. acres eligible for onions in stage i)
ui = vector of decision variables in stage i
(e.g. acres of onions to plant in stage i)
ri = random variable at stage i
ki = the value of ri
pi {ki } = probability that ri takes the value ki
. = stage return function
ai {}
. = state transformation function
ti {}
F { xn +1} = terminal value function
Important characteristics of the SDP include the objective function’s
recursive form and the expectation taken over future returns. The recursive
objective function represents the dynamic nature of the system being modeled.
Decisions made in stage i affect decisions in stages i+1 through i+n. This is
comparable to the dynamics captured by SP problems, in which first-stage
decisions (x) affect second-stage decisions (y) (note that an SP problem can have
more than two stages). The stochastic nature of the decision problem, reflected in
the objective function, is attributable to random variables that, in combination with
decision variables, determine returns and the state of the system in each stage.
Random variables in a SDP problem are not realized prior to the decision. Hence,
32
the objective function is the expected present value of returns from all stages.
Stochastic dynamic programming is a relatively straightforward extension of
dynamic programming, which is a common tool in economics; therefore, it will
not be discussed further.
A brief explanation of the difference between stochastic dynamic
programming and stochastic programming is warranted, however, because both
approaches can be used to model multi-stage decision-making under uncertainty.
Haneveld (1986, p. 2) suggests that while SDP and SP are similar in purpose, they
arose separately to address fundamentally different problems, and are not equally
suited for addressing the same problems. SDP stemmed from a need to costeffectively manage dynamic systems that were only partly controllable, often over
many stages. SDP is therefore best-suited to decision problems that involve few
(often discrete) decision variables and many stages.
SDP is a popular approach for forestry and fishery problems, for example,
in which the decision-maker chooses harvest levels throughout a long planning
horizon (often twenty stages or more). SP, in contrast, resulted from an effort to
incorporate random variables, such as price and yield, as parameters in LP
problems. LP models often involve many (often continuous) decision variables,
and relatively few stages; thus SP was developed to accommodate these features.
SP is a popular approach for farm management problems, for example, in which
the decision-maker chooses levels for a variety of farm activities throughout a
relatively short planning horizon (often three stages or less). Both approaches
unfortunately suffer the curse of dimensionality, and can therefore handle only a
limited number of random variables (Featherstone, Baker, and Preckel 1993; Toft
and O'Hanlon 1979).
Although SDP and SP are suited for problems with different
characteristics, some decision problems can be solved using either approach. In
particular, multi-stage SP models that have random variables with finite discrete
distributions can often be reformulated as discrete-time SDP models with a finite
33
number of stages (Haneveld 1986, p. 43). One advantage to using SDP rather than
SP is that analytical solutions can be derived for some problems, while only
numerical solutions are possible with SP. Kennedy (1986, p. 300) indicates,
however, that solutions to most SDP problems are determined numerically. In this
case, SP has the advantage of well-developed, commercially available solution
algorithms, whereas SDP models often require problem-specific algorithms. A
final distinction between the approaches is that SDP models typically assume a
Markovian structure, such that actions and outcomes depend only on the current
state of the system (Birge and Louveaux 1997, p. 70). That is, you do not need to
know how you arrived at your current state to determine the optimal next move.
SP, in contrast, can accommodate a variety of recursive structures.
2.3
Economic Studies of Agricultural Water Shortage
Many economic studies address water supply uncertainty and farm
management. Example objectives include reporting producers’ actual responses to
water supply uncertainty (Schuck, Frasier, and Webb 2003; Zilberman et al. 2002),
identifying optimal farm management in anticipation of, and response to, water
supply uncertainty (Bernardo et al. 1987; Wyse 2004), estimating the economic
impact of water supply uncertainty (Easterling 1993), estimating the value of
improved water supply forecasts (Mjelde, Hill, and Griffiths 1998; Mjelde,
Penson, and Nixon 2000; Wyse 2004), and estimating the ability of government
policies or water markets to reduce the impact of water supply uncertainty (Becker
1999; Burke, Adams, and Wallender 2004; Jaeger 2004).
Several studies have surveyed agricultural producers during or after a
drought to document their responses (Kromm and White 1986; Rich 1993; Schuck,
Frasier, and Webb 2003; Zilberman et al. 2002). These studies consistently
identify increased groundwater pumping and fallowing as primary responses to
drought, and deficit irrigation, improving irrigation efficiency, or adjusting crop
mixes as secondary responses. Each study solicits information from a diversity of
34
farm systems and producers, and is therefore able to identify common themes in
drought response. These studies provide an incomplete picture, however, of
drought preparedness and response. First, they do not report specific response
strategies for a particular farm system, or identify the degree to which alternative
drought response tools are used on individual farms. Second, they do not discuss
how producers prepare for drought in their year-to-year activities. Finally, they do
not analyze the optimality of observed drought responses.
Studies that use simulation or optimization models complement surveybased studies by providing insights about the optimality of alternative drought
preparedness and response tools for individual farm systems and producers. In
contrast to survey-based studies, however, it is challenging to draw general
conclusions from the model-based studies. This is because they span a wide
variety of farm systems, adopt different scales of time and space, focus on
different sets of drought management tools, and make a variety of assumptions
about uncertainty and dynamics. Some relevant studies are summarized below to
illustrate the diversity of methods, objectives, settings, and conclusions in this
body of literature.
Ziari and McCarl (1995) use a two-stage single-year stochastic
programming model to determine that a runoff collection impoundment is an
economical source of supplemental irrigation for mixed crop producers in Texas.
Iglesias et al. (2003) solves a multi-year dynamic model for several farm systems
in Spain, and concludes that improvements in the inter-year management of
reservoir levels and perfect water supply forecasts could mitigate the impacts of a
multi-year drought. Bernardo et al. (1987) constructs a single-year model that
captures intra-year irrigation dynamics, and applies it to a representative farm in
Washington’s Columbia River Basin. This study finds that a surface irrigator who
anticipates a water shortage should increase labor to improve irrigation efficiency,
decrease the frequency and depth of individual water applications, and deficit
irrigate crops during non-critical growth stages. Mejias et al. (2004), using a
35
three-stage single-year stochastic programming model, finds that a mixed crop
producer in southern Spain should shift to crops with lower water requirements in
response to increases in the water price (related to water scarcity).
Tapp et al. (1998) uses a five-year stochastic budgeting model to simulate
the ability of various financial and herd management strategies to mitigate the
impacts of drought for a livestock system in Australia. They conclude that no
strategy clearly dominates the others, and that no strategy successfully mitigates
the impacts of sustained drought. Toft and O’Hanlon also consider livestock
management in Australia during drought, but use an 18-month stochastic dynamic
programming model, rather than simulation, and focus on herd management
options only.
Kaiser et al. (1993), using a two-stage stochastic programming model,
finds that a corn-soybean producer in the Midwest should make few changes to
their crop mix in response to more frequent drought. Easterling (1993) also
focuses on the response of crop producers in the Midwest to short-term versus
sustained drought. They conclude from a multi-year simulation model that the
most effective adjustments include shifting planting dates, selecting longer-season
cultivars, and using furrow-dikes to capture rainfall. Finally, Weisensel et al.
(1991) develop a multi-year stochastic dynamic model for dryland wheat
production in western Canada. The authors compare expected net return and
variance of return for a fixed wheat-fallow rotation versus a flexible rotation based
on available soil moisture data. The flexible strategy generates higher expected
net return, but higher variance of return as well.
The above studies reflect that the structure of a producer’s decision
problem and the set of relevant drought management tools vary across farm
systems. The results for one farm system should therefore not be expected to
transfer directly to another. The modeling methods, however, are transferable;
which method is most appropriate for a chosen farm system depends largely on the
chosen assumptions about uncertainty and dynamics. Four alternative sets of
36
assumptions are common: 1) certainty with no dynamics (Adams and Cho 1998;
Bernardo et al. 1987; Jaeger 2004; Michelsen and Young 1993), 2) certainty with
intra- or inter-year dynamics (Garrido and Gomez-Ramos 2000; Haouari and
Azaiez 2001; Iglesias, Garrido, and Gomez-Ramos 2003; Thompson and Powell
1998), 3) uncertainty with intra-year dynamics (Adams et al. 1995; Kaiser et al.
1993; Keplinger et al. 1998; Mejias, Varela-Ortega, and Flichman 2004; Taylor
and Young 1995; Turner and Perry 1997), and 4) uncertainty with inter-year
dynamics (and in some cases intra-year dynamics as well) (Monke 1995; Toft and
O'Hanlon 1979; Weisensel, Van Kooten, and Schoney 1991).
The fourth set of assumptions most thoroughly captures the decisionmaking environment of a producer who faces uncertainty and a dynamic farm
system. Few studies have used this set of assumptions, however, to address
optimal drought preparedness and response (Antle 1983). No studies, to the
author’s knowledge, have used this set of assumptions in the context of an
irrigated crop farm (the farm system of interest in this dissertation). Some farm
systems may not involve all components of this set of assumptions, or researchers
may not be interested in all components. An alternative explanation is that models
that capture uncertainty and both intra- and inter-year dynamics are analytically
intractable and often difficult to solve numerically. However, computer
technology has advanced to the point that very large models can be solved within
acceptable time limits. The role of inter-year dynamics in optimal drought
preparedness and response can therefore be examined in the presence of other
important characteristics of the decision environment.
2.4
The Multi-peril Crop Insurance Program’s Prevented Planting Provision
The federal government actively assists agricultural producers with the
management of risk, and the mitigation of severe events, such as drought. The
USDA-Risk Management Agency’s goal, for example, is to “promote, support and
regulate sound risk management solutions to preserve and strengthen the economic
37
stability of America's agricultural producers” (Risk Management Agency 2004b).
The United States Department of Agriculture (USDA)-Farm Service Agency’s
goal is “to provide a safety net to help farmers produce an adequate food supply,
maintain viable operations, compete for export sales of commodities in the world
marketplace, and contribute to the year-round availability of a variety of low-cost,
safe, and nutritious foods” (Farm Service Agency 2002).
These and other federal agencies have shown particular interest in the
potential for crop insurance products to reduce economic losses associated with
drought. Data suggests that existing crop insurance programs have provided
significant financial support during recent drought events. The USDA’s Federal
Crop Insurance Corporation has, since 1989, paid insured producers a total of $462
million annually, on average, for qualifying drought losses (Office of
Communications 2004). In 2003 and 2002, fifty-four percent of the $3.2 billion,
and 60% of the $4.1 billion in total crop insurance indemnities paid, respectively,
were attributable to drought (Office of Communications 2004). Although these
transfers generally benefit the individual agricultural producers that receive them,
there is much debate about the implications of these wealth transfers and other
forms of government intervention for social efficiency. This aspect of farm policy
is not reviewed here; however, the economics literature offers a rich discourse on
the subject (e.g. Alston and Hurd 1990; Leathers and Chavas 1986; Luttrell 1989;
Pasour and Rucker 2005).
Interest in crop insurance is reflected in recent government policies. Title
X of the Farm Security and Rural Investment Act of 2002 (commonly referred to
as the “2002 Farm Bill”), for example, includes provisions to expand existing crop
insurance programs to include more crop types and locations (107th United States
Congress 2002). Section 10108 initiated a feasibility study of expanding coverage
to include disaster conditions caused by federal actions that restrict access to
irrigation water. The potential for disruption of irrigation water supplies by federal
actions increases as the list of threatened or endangered species expands. The roles
38
for such a program will likely change through time, particularly as the effects of
climate change materialize.
Research on crop insurance is robust, and includes explanatory models of
crop insurance participation (Calvin 1992; Just, Calvin, and Quiggin 1999;
Leathers 1994; Mahul 1999; Makki and Somwaru 2001; Sherrick et al. 2004),
program design (Mahul 1999; Makki and Somwaru 2001), production and market
effects of subsidized crop insurance (Glauber and Collins 2002; Hueth and Furtan
1994; Wu and Adams 2001; Young, Vandeveer, and Schnepf 2001), and the
interaction or substitutability of crop insurance with other risk management tools,
such as improved forecasts and disaster assistance (King and Oamek 1983; Luo,
Skees, and Marchant 1994; Mjelde, Thompson, and Nixon 1996). The role of crop
insurance, specifically the multi-peril crop insurance (MPCI) program’s prevented
plantings provision, as a drought preparedness tool is of interest here.
The USDA Risk Management Agency (RMA) has developed many
insurance products to help producers better manage production and price risks.
Provisions for crop insurance products are made in Title X of the Farm Bill. The
federal government subsidizes many of the insurance products to encourage
participation. Several insurance products, which are sold to producers through
private insurance companies, are available to producers in the study area (table
2.2). Crop insurance became available in Malheur County (the county in which
the study area is located) in 1990. Over $7.5 million has been paid in crop
insurance indemnities to producers in the county since then. Onions, sugar beets,
and wheat account for the largest portion of these indemnity payments. Twentyeight percent of insurance claims are attributed to drought or failure of irrigation
supply.
Prevented planting (PP), a provision included in basic MPCI policies for
irrigated crops, is becoming an increasingly popular form of drought preparedness
for producers in the study area (Agricultural Producers in the Vale Oregon
Irrigation District 2003; Haight 2004). A PP payment is made when a producer
39
provides evidence that as of the final planting date they had no reasonable
expectation of receiving sufficient water to follow standard irrigation practices,
due to an insurable cause of loss, such as drought (in contrast to an uninsurable
cause of loss, such as infrastructure failure). The cause of loss must also affect the
surrounding area and prevent other producers from planting acreage with similar
characteristics (Risk Management Agency 2003). This contrasts to a traditional
MPCI claim, where a crop was planted, but later failed due to unanticipated
drought.
Table 2.2. Crop insurance programs available, and eligible crops, in the
study area (Risk Management Agency 2004a).
Insurance
Program
MPCI
Multi-peril Crop
Insurance
Eligible Crops
alfalfa seed, apples, barley, corn, dry beans,
forage production, oats, onions, potatoes,
processing beans, sugar beets, wheat
CRC
Crop Revenue Coverage
corn, wheat
AGR
Adjusted Gross Revenue
all crops
IP
Income Protection
barley, wheat
The PP provision encourages producers to avoid planting when the crop is
expected to fail. Producers forego spring planting costs when they expect to make
a PP claim; therefore, the PP loss payment is typically a fraction of the normal
MPCI payment. The percentage is set at a starting level (e.g. 45% for onions and
sugar beets, 25% for potatoes, and 60% for wheat), and additional PP coverage can
be purchased for some crops (Haight 2004). A PP payment is calculated as
[approved yield * coverage level election* price election * share in crop * PP
40
percentage]. Note that the first four terms in the brackets determine the normal
MPCI payment. Producers cannot plant a substitute crop on the acreage submitted
for a PP claim without forfeiting the payment (Haight 2004). Producers are
allowed to plant some cover crops without forfeiting their PP payment, if the crop
is not harvested for grain or seed and is not a normal part of a rotation program.
Lastly, producers cannot rent acreage to other users if it has been submitted for a
PP claim (Haight 2004). Producers in the study area indicate that the MPCI
program’s prevented planting provision is a useful drought preparedness tool.
However, to the author’s knowledge, no economic studies have examined the
prevented planting provision in this role. Existing studies have focused instead on
the provision’s susceptibility to adverse selection and fraudulent claims (Rejesus,
Escalante, and Lovell 2005; Rejesus et al. 2003).
2.5
Climate Change and Drought in the Western United States
The ability of the earth’s atmosphere to trap solar radiation and increase
global temperature (the so-called “greenhouse effect”) has been recognized for at
least 150 years. More recently, global climate change has been a topic of intense
scientific and political debate. Certain evidence is unequivocal; carbon dioxide
concentrations (the most abundant greenhouse gas in the earth’s atmosphere) have
been increasing steadily for over a century. Specifically, CO2 levels have
increased 30% since the late 1800s, and are higher now than they have been in the
last 400,000 years (National Assessment Synthesis Team 2000). The decade of
the 1990s was also the warmest (on a global scale) in over a century. Average
annual temperature of the United States has risen almost 0.6° C (1.0° F) over the
20th century (National Assessment Synthesis Team 2000). The role that humans
have played in recent global warming, and whether it is possible to offset that
effect in any meaningful time scale, is still debated. The belief that global
warming will continue, however, is becoming more widely accepted in the science
41
and policy communities. It is prudent, therefore, to consider the impacts of such
warming on the frequency and severity of drought in the western United States.
Several general circulation models (GCMs) have predicted U.S. average
annual temperatures to increase 3 to 5° C (5 to 9° F) over the next 100 years
(National Assessment Synthesis Team 2000). Atmospheric scientists anticipate
numerous climatic effects to arise from these increasing temperatures. For
example, precipitation, which has increased in the U.S. by 5 to 10% over the 20th
century (Intergovernmental Panel on Climate Change 2001b), is predicted to
continue to increase in many regions, particularly those at higher
latitudes(Frederick and Gleick 1999; Gleick 2000). Two GCMs, the Canadian
Climate Centre, and the Hadley Centre in the United Kingdom, have projected
specific precipitation changes across the U.S. These include 25% precipitation
increases in the Northeast, 10 to 30% increases in the Midwest, 20% increases in
the Pacific Northwest, 10% precipitation decreases in the southern coast of Alaska,
and up to 25% declines in the Oklahoma panhandle, north Texas, eastern Colorado
and western Kansas (National Assessment Synthesis Team 2000). Caution should
be exercised in using any of these as predictions, given the coarseness of
geographical scale in existing GCMs.
Increases in precipitation, given warmer atmospheric conditions, will not
necessarily mean more available water at the state or regional level. The higher
evaporation rates that accompany rising temperatures are expected to result in less
water available in many regions (Frederick and Gleick 1999). For example,
GCMs project global average evaporation to increase 3 to 15% with doubled CO2
levels (Gleick 2000). Simulation studies suggest that precipitation must increase
by at least 10% to balance evaporative losses resulting from a 4° C temperature
increase(Gleick 2000). Projections of rising evaporation rates indicate they will
outpace precipitation increases, on a seasonal basis, in many regions (Gleick 2000;
Intergovernmental Panel on Climate Change 1998). The greatest deficits are
expected to occur in the summer, leading to decreased soil moisture levels and
42
more frequent and severe agricultural drought (Gleick 2000; Intergovernmental
Panel on Climate Change 1998).
Shifts in the form and timing of precipitation and runoff, specifically in
snow-fed basins, are also likely to cause more frequent summer droughts. More
precisely, rising temperatures are expected to increase the proportion of winter
precipitation received as rain, with a declining proportion arriving in the form of
snow (Frederick and Gleick 1999; Intergovernmental Panel on Climate Change
2001a). It is expected that snow pack levels will form much later in the winter,
accumulate in much smaller quantities, and melt earlier in the season
(Intergovernmental Panel on Climate Change 2001a).
These changes in snow pack and runoff are of particular concern to
irrigated agriculture. For example, if the runoff season occurs primarily in winter
and early spring, rather than late spring and summer, water availability for
summer-irrigated crops might decline during crucial spring and summer months,
causing water shortages to occur earlier in the growing season. Shifts in runoff,
precipitation and evaporation patterns may also intensify interstate and
international water allocation conflicts, as water managers struggle to meet
obligations of compacts and court decrees given more variable water availability
and timing in headwater areas. Global climate change is clearly relevant to
drought in agriculture. The effect of more frequent and severe drought on farm
income and optimal crop plans is investigated in chapter 5.
43
3
3.1
Description of the Study Area
Overview
The Vale Oregon Irrigation District (VOID) is the study area chosen for
this research. VOID includes 35,000 acres of irrigable lands encompassing the
towns of Harper, Little Valley, Vale, Willow Creek, and Jamieson, in Malheur
County, Oregon. Vale, Oregon, with a population of 1,976, elevation of 2,244
feet, and average annual precipitation of 9.77 inches is the largest and most central
town to VOID (Malheur County Oregon 2003).
3.2
Reservoirs
The Vale Oregon Irrigation District is located along the Malheur River,
Willow Creek, and Bully Creek drainages in northeastern Malheur County,
Oregon (figure 3.1). Neighboring irrigation districts include the Warmsprings
Irrigation District, Owyhee Irrigation District, and Orchards Water Company
(figure 3.2). Settlers began irrigating lands now included in VOID in 1881
(Bureau of Reclamation 1998b), which established the 1881 priority date for most
of VOID’s reservoir storage rights. The Vale Project was funded in 1926 through
an agreement between VOID and the federal government. In addition to
purchasing one-half of the storage rights to the existing Warmsprings Reservoir
from the neighboring Warmsprings Irrigation District, a diversion dam, main
canal, and lateral canals were to be built. The first unit of the Vale Project was
open for irrigation in 1930 (Bureau of Reclamation 1998b). Two additional
reservoirs were built for VOID, in 1935 and 1963.
44
Figure 3.1. The Malheur River Basin, including Bully Creek Reservoir,
Agency (Beulah) Reservoir, and Warmsprings Reservoir, which serve the
Vale Oregon Irrigation District (adapted from Shock et al. 2001).
45
Figure 3.2. Vale Oregon Irrigation District and neighboring irrigation
districts, Malheur County, Oregon (Unknown 19--).
46
Storage rights in Warmsprings, Beulah, and Bully Creek Reservoirs, and
surface rights from the Malheur River, Willow Creek, and Bully Creek provide
water for VOID (figure 3.1). The Bureau of Reclamation owns each of the
reservoirs. The Warmsprings Dam and Reservoir, located on the Middle Fork of
the Malheur River, 60 miles west of Vale, Oregon, collects spring snowmelt and
runoff from a drainage area of 1,110 square miles, and has an active storage
capacity of 191,000 acre-feet. VOID owns one-half, or 95,000 acre-feet, of this
storage capacity. The Warmsprings Irrigation District owns the other one-half of
the storage capacity and operates the dam on behalf of both irrigation districts
(Bureau of Reclamation 1998b). The Agency Valley Dam and Beulah Reservoir,
built in 1935, are located on the North Fork of the Malheur River, 18 miles north
of Juntura, Oregon. Beulah Reservoir collects snowmelt and runoff from a
drainage area of 444 square miles, and has an active storage capacity of 59,900
acre-feet. VOID operates the dam (Bureau of Reclamation 1998b). The Bully
Creek Dam and Reservoir, built in 1963, are located on Bully Creek, 9 miles
northwest of Vale, Oregon. Bully Creek Reservoir collects snowmelt and runoff
from a drainage area of 550 square miles, and has an active storage capacity of
30,000 acre-feet. VOID operates the dam (Bureau of Reclamation 1998b). In
addition to storing Bully Creek flows, surplus winter flows in the Malheur River
are also diverted to and stored in Bully Creek Reservoir for use the following
growing season.
Total reservoir storage capacity available to VOID is 184,900 acre-feet.
Water is drawn from a total drainage area of 2,104 square miles to irrigate 35,000
acres. In comparison, the neighboring Owyhee Irrigation District receives its
irrigation water from Owyhee Reservoir, which has an active storage capacity of
715,200 acre-feet, and collects water from a drainage area of 10,900 square miles
(Bureau of Reclamation 1998a). Owyhee Reservoir serves 105,000 irrigated acres,
including the Owyhee Irrigation District’s 65,000 irrigated acres (Jacobs 2003).
47
VOID can store enough water to irrigate for 1.5 seasons provided that
Warmsprings, Beulah, and Bully Creek Reservoirs fill to capacity (Ward 2004).
To estimate how frequently these reservoirs are filled, the fraction of water years
over a 35-year period during which the ‘maximum reservoir content’ exceeded
85% of the reservoir’s storage capacity was calculated (Bureau of Reclamation
2004). Bully Creek Reservoir content exceeded 85% of the storage capacity
during 25 out of 35 years (71.4%). Content was less than 50% of storage capacity
in the years 1988, 1991, and 1992. Beulah Reservoir content exceeded 85% of
storage capacity during 23 out of 35 years (65.7%). Content was less than 50% of
storage capacity in the years 1988, 1991, and 1992. Warmsprings Reservoir
content exceeded 85% of storage capacity during 22 out of 35 years (62.9%).
During the years 1988, 1991, and 1992 Warmsprings’ maximum reservoir content
did not exceed 25% of storage capacity. Warmsprings Reservoir also remained at
less than 50% of storage capacity in the years 1997, and 2001 to 2004. Beulah and
Warmsprings Reservoirs are located only 30 miles apart; therefore, they often
suffer simultaneous shortages. Table 3.1 reports the frequency of ‘percent
reservoir storage capacity’ classes over the 35-year period.
Water stored in Warmsprings and Beulah Reservoirs must travel 65 miles
east via the Malheur River before it reaches VOID’s main diversion, the Harper
Diversion Dam. Water is diverted there into the earthen Vale Main Canal, through
which it must travel 74 miles along the western border of VOID to reach the end
of the canal, near Jamieson, Oregon. The canal empties there into Willow Creek,
which acts as a conduit for producers in the Willow Creek area. As water moves
through the Vale Main Canal, it is diverted into numerous lateral canals and
pipelines for delivery to individual VOID producers (Bureau of Reclamation
1998a). Water stored in Bully Creek Reservoir is released into Bully Creek and
diverted to nearby VOID acreage. An extensive drainage system is also in place
throughout VOID to manage surface and subsurface runoff, which play an
48
important role in water allocation within VOID and between VOID and the
Warmsprings Irrigation District (Bureau of Reclamation 1998a).
Table 3.1. Count and percent occurrence of five reservoir storage level
classes during water years 1969-2004 at Bully Creek, Beulah, and
Warmsprings Reservoirs (Bureau of Reclamation 2004).
Bully Creek Reservoir Storage Capacity (acft): 31,650
Acre-feet
% of storage capacity
# years
% occurrence
0 - 7,913
0-25%
0
0.0%
7,914 - 15,825
25.1-50%
3
8.6%
15,826 - 23,738
50.1-75%
5
14.3%
23,739 - 26,903
75.1-85%
2
5.7%
26,904 - 31,650
85.1% +
25
71.4%
Storage Capacity (acft): 59,900
Beulah Reservoir
Acre-feet
% of storage capacity
# years
% occurrence
0 - 14,975
0-25%
0
0.0%
14,976 - 29,950
25.1-50%
3
8.6%
29,951 - 44,925
50.1-75%
6
17.1%
44,926 - 50,915
75.1-85%
3
8.6%
50,916 - 59,900
85.1% +
23
65.7%
Warmsprings Reservoir Storage Capacity (acft): 191,100
Acre-feet
% of storage capacity
# years
% occurrence
0 - 47,775
0-25%
3
8.6%
47,776 - 95,550
25.1-50%
5
14.3%
95,551 - 143,325
50.1-75%
4
11.4%
143,326 - 162,435
75.1-85%
1
2.9%
162,436 - 191,100
85.1% +
22
62.9%
49
3.3
The Distribution of Water
VOID appropriates an equal per acre water allotment to each district
member. The appropriation amount is determined at the beginning of each season,
and is based on many factors, including current and expected reservoir levels.
Recall that all VOID water rights share the same 1881 priority date, thus all
members have equal rights. Producers are allowed to apply their allotment to any
of their acreage, but cannot easily transfer water to another VOID member (Jacobs
2003). Producers are assessed a fee of $80 per account plus $29 per acre, per year,
regardless of the quantity of water delivered (Vale Oregon Irrigation District
2004a).
A full allotment in VOID is 3.5 acre-feet of water per acre of land (Ward
2004), although they are allowed up to 4.5 acre-feet per acre in an unrestricted
water year (Jacobs 2003). Since 1992, VOID members have received a full
allotment only once, in 1997. Allotment hit a record low of 0.92 acre-feet per acre
in 1992 (Vale Oregon Irrigation District 2004b). Table 3.2 reports district
allotments for years 1981 to 2003 (Vale Oregon Irrigation District 2004b).
Interviews with producers suggest that most crops can be grown successfully with
a 3 acre-feet per acre allotment. Producers received 3 or more acre-feet per acre
during 11 of 23 years (48%) for the period 1981 to 2003.
Some VOID members have access to supplemental water sources.
Members with canal access to Bully Creek Reservoir sometimes receive a larger
allotment, because distribution from this reservoir cannot be distributed to all
VOID members. A small number of producers have individual stream-flow rights
to Malheur River, Willow Creek, or Bully Creek, or have access to groundwater.
Twenty groundwater wells were in place at the end of the 2003 growing season
(Jacobs 2003). The number of economically feasible wells is limited in VOID,
due in part to deep aquifers (400 to 600 feet or more) and high electricity costs, but
producers continue to search for shallow aquifers (Jacobs 2003).
50
Table 3.2. VOID water allotments (acre-feet per acre) for the years 19812003 (Vale Oregon Irrigation District 2004b).
Year
2003
2002
2001
2000
1999
1998
1997
1996
1995
1994
1993
1992
3.4
Acre-feet
per acre
1.75
2.20
2.13
3.20
3.03
2.74
3.56
3.02
2.71
2.54
3.05
0.92
Year
1991
1990
1989
1988
1987
1986
1985
1984
1983
1982
1981
Acre-feet
per acre
1.06
2.05
2.90
1.02
2.89
3.55
3.46
3.60
3.65
3.58
3.60
Agricultural Production
Today, up to 35,000 acres of land are irrigated for agricultural purposes
during the best water years. The eleven-year average of irrigated acreage for the
years 1992 to 2002, including acreage irrigated but not harvested (an average of
570 acres), is 33,830 acres. The eleven-year average acreage fallowed or idled is
1,000 acres. During the extremely dry year of 1992, irrigated acreage fell to
28,100 acres, with nearly 6,800 acres fallowed. Records indicate that 216 fulltime farms and 207 part-time farms operated within VOID in the year 2000,
compared to 230 full-time farms and 180 part-time farms that operated within
VOID in the year 1992 (Vale Oregon Irrigation District 2004b).
Soil quality in VOID is categorized into bench-land (Frohman-Virtue) and
bottom-land (Powder-Turbyfill-Garbutt) (Soil Conservation Service and Oregon
Agricultural Experiment Station 1979). Bottom-land has higher quality soil
compared to bench-land, because it is flatter and deeper, which increases soil
moisture retention. The highest quality land in VOID is a one-mile wide strip of
51
bottom-land that buffers Willow Creek. Land west of Willow Creek, outside of
the buffer area is bench-land, which tends to be hilly and have shallow soils. A
similar pattern occurs along Bully Creek. Land surrounding the town of Harper
has deeper soils, but it is hillier than most bottom-land. Most remaining land in
VOID is bench-land (Soil Conservation Service and Oregon Agricultural
Experiment Station 1979).
It is important to note that the best land in VOID occurs along its streams,
and that farms located on bottom-land along streams have a higher probability of
owning stream rights. This combination of having the best soils in VOID and
owning supplemental water rights is likely an advantage during periods of drought.
Recall, however, that land along Willow Creek is located at the end of the Vale
Main Canal, making it more difficult to deliver water to the area during a drought,
or at the end of the irrigation season.
Alfalfa hay and irrigated pasture account for roughly 57% of VOID’s
irrigated acreage. Wheat, corn, and forage for silage account for 24% of irrigated
acreage. Other hay, barley, sugarbeets, dry beans, potatoes, and onions account
for 16% of irrigated acreage (Vale Oregon Irrigation District 2004b). Table 3.3
reports ‘harvested acreage by crop’ as a percent of total harvested acreage for the
period 1992 to 2002.
3.5
Irrigation Technology
Most of VOID’s canals, including the Vale Main Canal, are unlined.
However, canals are slowly being replaced with pipelines in an effort to reduce
conveyance losses. Currently, over 130,000 feet of pipeline has been installed
(Ward 2004), and 36,000 feet of additional pipeline is being installed to reduce
Escherichia coli levels in Willow Creek. Furrow irrigation, using siphon tubes or
gated pipes, is the predominant irrigation technology in the VOID. However,
acreage under sprinkler irrigation (solid set, wheel line, and center pivot) grew
from 4,000 acres (11.4% of district acreage) in 1992 to 5,500 acres (15.7%) in
52
2001 (Vale Oregon Irrigation District 2004b). The installation of additional
sprinkler irrigation systems is limited, however, because most of VOID’s bottomland is too flat to generate sufficient head pressure, and thus electric pumps are
required to pressurize the system (Ward 2004). Producers indicate that the cost of
the infrastructure required to deliver electricity to the pressurized sprinkler systems
is prohibitive.
VOID onion growers have begun adopting drip irrigation. One-hundred
acres were under drip irrigation in 2002, and as of 2004 this acreage has doubled
(Vale Oregon Irrigation District 2004b). Drip irrigation is used exclusively on
onions in VOID because no other crop is sufficiently valuable to justify the
installation and operating expenses. VOID producers are also installing reuse
furrow systems to improve on-farm irrigation efficiency. Collection ponds are dug
at the bottom of fields to collect surface and subsurface runoff. Pumps, filters, and
pipelines then deliver the recycled water to the same field or to neighboring fields.
Reuse systems reduce and alter return flows, often to the detriment of downstream
producers. Warmsprings Irrigation District relies heavily on VOID’s return flows,
and have already reported impacts from such water conservation efforts (Ward
2004). Producers who choose not to invest in improved irrigation technology
combat water shortages by allocating more labor to the management of their
furrow irrigation system. However, labor is difficult to find during the irrigation
season, so producers must weigh the cost of additional labor against the benefit of
improved irrigation efficiency.
3.6
Water Supply Forecasts
Snowpack is the primary source of irrigation water to VOID. The
transformation of winter snowpack to spring runoff is a complex process, and a
major source of uncertainty for VOID managers and producers. Water supply
forecasts for the VOID currently have a limited forecast horizon (predictions are
typically formulated no earlier than April), and limited accuracy. An exploratory
53
economic analysis indicates positive economic benefit to VOID from improved
water supply forecasts (Wyse 2004).
3.7
Summary
The Vale Oregon Irrigation District has many characteristics that make it
an appropriate empirical focus for this research: 1) VOID producers experience
drought frequently (most recently a three-year drought that ended in 2004), which
has resulted in application of numerous drought preparedness and response tools;
2) row-crop systems in the VOID involve both intra- and inter-year dynamics, and
3) producers have indicated a desire to enhance their ability to prepare for, and
respond to drought. The model is therefore parameterized for a hypothetical
irrigated row-crop farm in VOID.
54
Table 3.3. Harvested acres by crop as a percent of total harvested acreage, 1992-2002 (Vale Oregon Irrigation District
2004b).
Year
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
Harvested Water Delivered
Acres
(Acft Per Acre) Alfalfa Hay Irrig. Pasture
28,136
0.92
28.0%
22.9%
33,079
3.05
25.9
23.7
33,753
2.54
29.4
25.5
33,589
2.71
29.9
25.7
34,376
3.02
26.0
26.9
34,345
3.56
27.2
26.4
34,467
2.74
32.8
26.9
33,850
3.03
33.0
28.5
33,839
3.20
33.8
29.3
34,410
2.13
38.6
29.3
33,816
2.20
42.6
27.2
Wheat
13.2%
9.8
9.7
9.8
15.4
15.7
10.0
6.1
6.0
4.9
4.5
Corn
2.3%
7.2
5.1
8.1
10.7
10.7
8.1
10.3
8.5
6.7
5.8
Silage
10.5%
10.3
8.9
7.6
5.9
5.9
6.7
5.5
5.6
5.0
6.5
54
55
Table 3.3. (Cont.)
Year
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
Other Hay
4.6%
9.0%
4.6%
4.6%
2.9%
2.6%
3.9%
3.7%
6.0%
7.5%
6.2%
Barley
(feed)
10.3%
4.8%
5.2%
4.2%
2.7%
2.7%
2.2%
2.3%
2.9%
1.5%
1.1%
Sugar beet
2.9%
2.1%
2.3%
2.6%
1.9%
1.4%
2.4%
2.0%
2.6%
2.6%
2.1%
Beans
(dry edible)
0.73%
2.55%
2.66%
2.61%
1.51%
1.53%
1.49%
2.10%
0.59%
0.24%
0.71%
Onions
(dry)
0.99%
1.24%
1.33%
1.12%
1.03%
1.01%
0.68%
1.29%
1.16%
1.34%
1.04%
Potatoes
(early)
1.14%
1.54%
1.68%
1.32%
1.71%
1.60%
1.21%
1.10%
0.67%
0.34%
0.00%
Potatoes
(late)
0.40%
0.75%
1.17%
1.31%
1.64%
1.54%
1.70%
1.82%
1.45%
1.53%
1.90%
55
56
4
4.1
Model Description
Overview of the Model
Stochastic programming (SP) is selected to represent the decision-process
of farm managers in the study area. SP is able to represent farm managers who
make fall decisions about several cropping activities, given an uncertain future
water supply, with explicit consideration of future spring decisions. Managers
decide in the fall the number of acres to plant to winter wheat, and the number of
acres to prepare for onions, potatoes, and sugar beets. The phenology of winter
wheat requires it to be fall-planted, and spring labor constraints require the fall
preparation of other acreage. Ideally, fall decisions would be made given perfect
knowledge of the upcoming growing season’s water supply. Unfortunately, only
the subjective probability distribution of the future water supply is known when
fall decisions are made. Fall decisions therefore represent the first stage of the SP
model.
Information enters the SP model decision framework in early spring when
a water supply forecast becomes available. The forecast is assumed to perfectly
predict the growing season’s water supply; this is not the case in reality, but is a
simplifying assumption. Upon receiving these forecasts, and subject to constraints
associated with their fall decisions, managers make their spring decisions. These
spring decisions represent the second stage of the SP model. Managers have some
recourse actions available at this time. Suppose, for example, that the spring
forecast indicates a full water allotment. A manager who left a large acreage open
in the previous fall could plant it to a spring-prepared and planted crop, such as
corn. SP would capture the structure of this decision process much better than
CCP or PP. SP is chosen over SDP because it accommodates several decision
variables, and because the desired number of stages is relatively small (twelve or
fewer).
57
Additional recourse stages exist in the manager’s true decision process.
These additional stages occur because the spring water supply forecast is
imperfect, and as the season progresses the manager gains additional information
about the seasonal water supply and responds by adjusting management activities,
such as irrigation and date of harvest. In fact, the actual value of the seasonal
water supply is not fully revealed until all crops have matured, since the water
supply includes rainfall events that cannot be predicted beyond a few days notice.
There are also other random variables that are not known a priori and that affect
managers’ optimal decision path, such as output prices and the occurrence of pests,
hail, and wind.
Dozens of stages could be constructed in to simulate farm managers’
decisions and flow of information over a year. Such an attempt is beyond the
scope of this research. The number of stages within a crop year is limited to two
(three for models that incorporate price uncertainty), and the number of random
variables will generally be limited to one, water supply; price is also treated as a
random variable in some models. Limiting the number of stages per year and the
number of random variables allows us to expand a single-year (two-stage) model
to a six-year (twelve-stage) model. Six years is chosen because the dynamic effect
of a specific crop in the study area spans a six-year period. The model is described
in detail below.
Many studies that use mathematical programming models represent farm
decisions as continuous variables (e.g. acres planted). A continuous variables
model was initially developed for this study to address questions about drought in
a multi-year context. One advantage of continuous variables models is that they
require a less powerful solver and less computational time. However, some interyear crop rotation constraints can be modeled only approximately within the
continuous variables specification (this is discussed in detail below). Concern
about the approximate nature of crop rotations in the continuous variables model
58
prompted the construction of an equivalent binary variables model. The
availability of both a continuous and binary variables model presents an
opportunity to illustrate the structure of each model, and compare their optimal
solutions. This comparison provides useful insights as to the differences in
outcomes between a binary and continuous model in a farm system context.
A general representation of the models is presented next, followed by an
explanation of the general features. A detailed presentation and description of
each model’s equations then follows. In the following discussions, the binary
model is typically presented first and in the most detail, followed by a brief
description of any features that are unique to the continuous model. Parameter
values used in the analyses are presented in the Appendices. All models are
constructed in GAMS (General Algebraic Modeling System) and solved using
CPLEX; the programming code is provided on an attached floppy disc.
4.1.1
(1)
The Binary Variables Model
Max Es Π ( x, y; s )
x, y
s.t.
(2)
Ax = b
(3)
Dy = e
(4)
(5)
Mx + Ny = g
x, y ≥ 0
where
s = A random vector that represents water supplies over a 6-year planning horizon.
Each realization of s consists of 6 components (s1 s2 s3 s4 s5 s6), which
indicate the state of nature (water supply category) revealed in each of the
six years. That is, s1 represents the state of nature revealed in year 1, s2 the
state of nature revealed in year 2, etc. Assuming 2 possible states of nature
(Dry or Full) in each of 6 years, 64 six-year water supply scenarios are
possible. Scenarios range from [Dry Dry Dry Dry Dry Dry] to [Full Full
59
Full Full Full Full], and every combination between. The scenario [Dry
Dry Full Full Full Full] indicates (from left to right) that the state of nature
revealed in year 1 is Dry, year 2 is Dry, year 3 is Full, etc. Each state of
nature has a probability of occurrence within any given year, denoted
pr(Dry) or pr(Full). The state of nature in any one year is assumed
independent of the state of nature in any other year (based on an
autocorrelation analysis of historical stream-flow above the storage
reservoirs as described by Haan (2002 p. 348). Therefore, the joint
probability of a particular six-year water scenario is the product of the
probabilities of the states of nature that occur each year. For example,
pr([Dry Dry Full Full Full Full]) = pr(Dry)*pr(Dry)*pr(Full)*
pr(Full)*pr(Full)*pr(Full). Historical water allotment data and Gaussian
quadrature analysis (Featherstone, Baker, and Preckel 1993; Miller and
Rice 1983; Preckel and Devuyst 1992) were used to assign quantity of
water and probability to each state of nature (Appendix B).
x = Vector containing fall crop decision variables for each year of the planning
horizon.
Example element: x3,f,c,i,s1,s2, which indicates for the fall of year 3, that field f
is prepared for or planted to crop c, under irrigation technology i, given the
states of nature revealed in past years 1 and 2. Each element of x is a
binary variable, taking on a value of 0 (if the crop/irrigation combination
(c,i) is not chosen for field f) or 1 (if the crop/irrigation combination (c,i) is
chosen for field f). Each field may also be left “open,” implying that it is
neither prepared for nor planted to any crop.
y = Vector of spring crop decision variables for each year of the planning horizon.
Example element: y3,f,c,i,w,s1,s2,s3, which indicates for the spring of year 3, that
field f is planted to crop c in the spring of year 3, under irrigation
technology i, and deficit irrigation category w, given the states of nature
60
revealed in past years 1, 2, and the present year 3. Each element of y is a
binary variable, taking on a value of 0 (if the crop-irrigation-deficit
combination (c,i,w) is not chosen for field f) or 1 (if the crop-irrigationdeficit combination (c,i,w) is chosen for field f). Each field may also be
“fallowed,” in which case it is either abandoned (if prepared or planted in
the previous fall), or simply never planted (if left open in the previous fall).
Π ( x, y; s ) = Vector containing the profit outcome for each water scenario. An
individual element of the vector is the discounted stream of profit that
optimal activities x and y generate over the 6-year period in which they
occur, for a particular water scenario. Terminal land rental values are also
included, as a function of activities in the 6-year period. A terminal value
for alfalfa acreage that remains in production after year 6 is also included.
A, D = Matrices of coefficients that describe fall and spring activities’ resource
use.
b, e = Vectors of resource availability, such as land and water, which vary by state
of nature for some resources.
M, N = Matrices of coefficients that relate activities in different time periods to
each other (intra- and inter-year rotation constraints).
g = Vector of parameters that, with M and N above, define relationships between
activities in different time periods.
4.1.2 The Continuous Variables Model
The continuous variables model, in its general form, is the same as the
binary variables model described above, except for the interpretation of the
decision variables. The new interpretation is provided below.
x = Vector containing fall crop decision variables for each year of the planning
horizon.
61
Example element: x3,c,i,s1,s2, which indicates for the fall of year 3 the number of
acres of crop c to prepare for or plant, under irrigation technology i, given
the states of nature revealed in years 1 and 2.
y = Vector of spring crop decision variables for each year of the planning horizon.
Example element: y3,c,i,w,s1,s2,s3, which indicates for the spring of year 3 the
number of acres of crop c to plant, under irrigation technology i, and
deficit irrigation category w, given the states of nature revealed in years 1,
2, and 3.
4.2
Interpreting the General Model
The above discrete stochastic sequential programming models (binary and
continuous) maximize the expected stream of profit over a 6-year planning
horizon. The expectation is taken over water supply, s, which is assumed to have a
discrete probability distribution over a small number of pre-defined categories
(e.g. dry and full). Choice variables are contained in the vectors x and y. Vector x
includes fall cropping activities, which are chosen under an uncertain future water
supply. Vector y includes spring cropping activities, which are chosen after water
supply is revealed. Fall and spring activities are chosen for each year of the sixyear planning horizon, for each water supply scenario, (e.g. [Full Full Full Dry Dry
Dry]). Fall and spring activities are constrained by resource availability, as
expressed in equations (2) and (3). Equation (4) describes dynamic interactions in
the cropping system, including how fall activities restrict spring activities (intrayear dynamics), and how activities in year t restrict activities in subsequent years
(inter-year dynamics).
The timing of decisions relative to the availability of water supply
information is an essential feature of the DSSP model (figure 4.1). Past water
supply is known in the base model, but future water supply is uncertain.
Specifically, fall cropping activities (xt) are chosen before the water supply for the
upcoming growing season is known, and before the water supplies for future
62
growing seasons are known. Water supply for the upcoming growing season is
revealed in early spring, after which spring cropping activities (yt) are chosen.
Note that although the water supply for the upcoming season is revealed in the
spring, the water supplies in future growing seasons remain uncertain. This
sequence of events (choose xt, water supply is revealed, choose yt) is repeated in
each year of the six-year planning horizon.
Intra- and inter-year dynamics between cropping activities require the
producer to be forward-looking to make optimal decisions. Activities in the fall of
year 1, for example, will constrain activities in the spring of year 1, which will
potentially constrain activities throughout the remainder of the planning horizon.
full
y1|x1
dry
y1|x1
x1
water
revealed
(spring)
profit outcome
p(wet)=0.6
p(dry)=0.4
water
uncertain
(fall)
Year 3
Year 2
x2|y1
x2|y1
water
uncertain
(fall)
full
y2|x2
dry
y2|x2
full
y2|x2
dry
y2|x2
x3|y2
profit outcome
Year 1
x3|y2
x3|y2
x3|y2
water
revealed
(spring)
Figure 4.1. Decision tree representation of decision-making under
uncertainty. Fall cropping activities (xt) are chosen given an uncertain
spring water supply. The spring water supply is then revealed (full or dry),
after which spring cropping activities (yt) are chosen.
Therefore, when selecting activities for fall of year 1, the producer must consider
the activities’ future impacts. Future impacts are challenging to identify, however,
63
because future water supplies are uncertain. The following example illustrates this
point.
Suppose, for simplicity, that the planning horizon is a single year, within
which fall decisions are made given an uncertain water supply, and spring
decisions are made given a certain water supply. When selecting fall activities, the
producer must consider the impact on spring activities for two cases: 1) the spring
allotment is revealed to be full, and 2) the spring is revealed to be dry. A
particular set of fall activities might maximize profit in the event of a full
allotment, but not a dry spring, or vice versa. In contrast, the optimal set of fall
activities will, by the definition of “optimal” in this dissertation, maximize
expected profit over both water scenarios. That is, the producer must select fall
activities based on their performance in each possible water scenario, and the
probability of each scenario.
A solution to this two-stage, single-year problem consists of one set of
optimal fall activities, and two sets of optimal spring activities, one for a full
spring allotment, and one for a dry spring. The producer implements the fall plan,
and after the water supply is revealed as full or dry, the producer implements the
corresponding spring plan. Suppose that the producer knows, prior to making their
fall decision, that the allotment will be full. The resulting set of optimal fall
activities would likely differ from the set derived under uncertainty. Note that the
difference in profit between these two scenarios represents the cost of uncertainty,
or equivalently, the value of perfect information.
When this model is expanded from one year to two, the producer identifies
one set of optimal “fall year 1” activities, two sets of optimal “spring year 1”
activities, two sets of optimal “fall year 2” activities, and four sets of optimal
“spring year 2” activities (figure 4.1). The producer, in choosing their activities
for fall year 1, considers that four water supply scenarios are possible over the
two-year period: [Full Full], [Full Dry], [Dry Full], and [Dry Dry]. In addition to
64
choosing a plan for fall year 1, the producer selects activities for each stage of
every possible water supply scenario. In reality, the producer will update year 2
plans once the outcome of year 1 is realized, in order to make full use of
information gained in year 1, and to look six years into the future before choosing
a year 2 plan. The plan made for year 2 in the fall of year 1 should therefore be
interpreted as an estimate of the optimal year 2 plan. Figure 4.1 illustrates the
branching pattern of crop plans that arises when future water supplies are
uncertain. Two states of nature and a six-year planning horizon are assumed in the
empirical model, which generates 64 unique water scenarios, and potentially 64
unique six-year crop plans. It is unlikely that a producer could envision six-year
crop plans for all sixty-four scenarios. It is likely, however, that a producer could
clearly envision crop plans and outcomes for the near future, as well as attempt to
consider the dynamic implications of those plans in the more distant future. The
farther into the future the producer can envision, the closer their year 1 plans will
be to the optimal.
Once a crop plan is determined for each stage (fall and spring) of each year
of each water supply scenario, a discounted stream of profit is calculated for each
scenario (i.e. for each branch of the decision tree). Expected profit over all
possible water supply scenarios is then calculated, given each scenario’s
probability of occurrence. The DSSP models are presented in detail next for both
the binary and continuous variables models.
4.3
Details of the Binary Variables Model
The equations of the binary variables model are presented below, followed
by their interpretation.
65
(6)
⎡ 6 ⎛ 1
⎤
⎞
π t , s ( xt , s , yt , s ) ⎟
⎢∑ ⎜
⎥
t
t =1 ⎝ (1 + d )
⎠
⎢
⎥
Max Es Π ( x, y; s ) = ∑ ρ s ⋅
⎢
12
x, y
⎛ 1
⎞⎥
s
⎢+∑ ⎜
⎥
π
y
y
(
,...
)
t ,s
t − 6, s
6, s ⎟
t
⎢⎣ t =7 ⎝ (1 + d )
⎠ ⎥⎦
where for t =1,...,6
⎛ ∑ FAC ⋅ ( pc yld c ,i , w yt , f ,c ,i , w, s ⎞
⎟
(7) π t , s ( xt , s , yt , s ) = ∑∑∑ ⎜ f
⎜
⎟
c
i
w
⎝ − jc ,i yt , f ,c ,i , w, s − hc ,i xt , f ,c ,i , s ) ⎠
⎛
⎛
⎞⎞
⎜ ∑∑∑ ⎜ ∑ FAC ⋅ ( jc ,i yt , f ,c ,i , w, s + hc ,i xt , f ,c ,i , s ) ⎟ ⎟
− fxd cos t − r ⋅ ⎜ c i w ⎝ f
⎠⎟
⎜ + fxd cos t − π ( x , y )
⎟
t −1, s
t −1, s
t −1, s
⎝
⎠
and
⎡ ⎛
⎛ w ⋅ ( ET max c − Ppt ) + Ppt ⎞ ⎞ ⎤
(8) yld c ,i , w = max yld c ,i ⋅ ⎢1 − ⎜⎜ kyc ⋅ ⎜ 1 −
⎟ ⎟⎟ ⎥
ET max c
⎝
⎠ ⎠ ⎦⎥
⎣⎢ ⎝
where for t =7,...,12
(9) π t , s ( yt −6,s ,... y6, s ) = RRateonion ⋅ EligOniont , s ( yt −6, s ,... y6, s )
+ RRateother ⋅ EligOthert , s ( yt −6, s ,... y6, s )
⎛
⎞
+ ⎜ ∑ NetRvalf ,i , d 1 ⋅ EligAlft ,i , s ( y6, s ) ⎟ − fxd cos t
⎝ i
⎠
⎛
⎞
− r ⋅ ⎜ ∑ ( jalf ,i ⋅ EligAlft ,i , s ( y6, s ) ) + fxd cos t − π t −1, s ( yt −6, s ,... y6, s ) ⎟
⎝ i
⎠
subject to
(10)
⎛ ET max c − Ppt ⎞
⎟ ⋅ FAC ⋅ yt , f ,c ,i , w, s
IrrEffici
⎝
⎠
∑∑∑∑ w ⋅ ⎜
c
i
w
f
≤ Watert , s ⋅ TotAcres
(11) xt , f ,c ,i , s = 0
(12) y t,f,c,i,w,s = 0
for some t,c,i,s ∀ f
for some t,c,i,w,s
(13) xt , f ,c ,i , s , yt , f ,c ,i , w, s = 0 or 1
(14)
∑x
1, f , wheat ,i
i
∀ t, s
+H 6 f , wheat + H 6 f ,barley ≤ 1
∀f
∀ t,f,c,i,w,s
∀f
66
(15)
∑x
+ H 6 f , sugbt + H 5 f , sugbt + H 4 f , sugbt
1, f , sugbt ,i
i
+ H 3 f , sugbt ≤ 1
(16)
∑x
∀f
+H 6 f ,onion + H 5 f ,onion + H 4 f ,onion
1, f , onion ,i
i
+ H 3 f ,onion + H 2 f ,onion ≤ 1
(17)
∑x
1, f , potato ,i
∀f
+H 6 f , potato + H 5 f , potato + H 4 f , potato
i
+ H 3 f , potato + H 2 f , potato ≤ 1
(18)
∑∑ x
1, f , potato ,i
i
(19)
∀f
≤ PotatoContract
f
∑x
= H 6 f ,alf 1
∀ f,i
∑x
= H 6 f ,alf 2
∀ f,i
∑x
= H 6 f ,alf 3
∀ f,i
1, f , alf 2,i
i
(20)
1, f , alf 3,i
i
(21)
1, f , alf 4,i
i
(22)
∑∑ x
1, f , c ,i
c
(23)
+ open f ,1 = 1
∀f
i
∑y
≤ x1, f , fall ,i
∀ f,fall,i,s
∑y
= x1, f ,alf 2,i
∀ f,i,s
∑y
= x1, f , alf 3,i
∀ f,i,s
∑y
= x1, f ,alf 4,i
∀ f,i,s
1, f , fall ,i , w , s
w
(24)
1, f , alf 2,i , w , s
w
(25)
1, f , alf 3,i , w, s
w
(26)
1, f , alf 4,i , w , s
w
(27)
∑∑ y
1, f , gcorn ,i , w , s
i
+ H 6 f , gcorn + H 6 f , scorn
w
+ H 5 f , gcorn + H 5 f , scorn ≤ 2
(28)
∑∑ y
1, f , scorn ,i , w , s
i
+ H 6 f , gcorn + H 6 f , scorn
w
+ H 5 f , gcorn + H 5 f , scorn ≤ 2
(29)
(30)
(31)
∑∑ y
1, f ,barley ,i , w, s
i
w
c
i
∑∑∑ y
1, f ,c ,i , w , s
∑x
w
2, f , wheat ,i , s
i
∀ f,s
+ H 6 f ,barley + H 6 f , wheat ≤ 1
∀ f,s
=1
∀ f,s
+ ∑∑ y1, f , wheat ,i , w, s + ∑∑ y1, f ,barley ,i , w, s ≤ 1
i
∀ f,s
w
i
w
∀ f,s
67
(32)
∑x
+ ∑∑ y1, f , sugbt ,i , w, s + H 6 f , sugbt
2, f , sugbt ,i , s
i
i
w
+ H 5 f , sugbt + H 4 f , sugbt ≤ 1
(33)
∑x
∀ f,s
+ ∑∑ y1, f ,onion ,i , w, s +H 6 f ,onion + H 5 f ,onion
2, f , onion ,i , s
i
i
w
+ H 4 f ,onion + H 3 f ,onion ≤ 1
(34)
∑x
2, f , potato ,i , s
+ ∑∑ y1, f , potato ,i , w, s +H 6 f , potato + H 5 f , potato
i
i
∀ f,s
w
+ H 4 f , potato + H 3 f , potato ≤ 1
(35)
∑∑ x
2, f , potato ,i , s
f
(36)
=∑∑ y1, f , alf 1,i , w, s
∀ f,s
∑x
=∑∑ y1, f ,alf 2,i , w, s
∀ f,s
∑x
=∑∑ y1, f , alf 3,i , w, s
∀ f,s
2, f , alf 2,i , s
2, f , alf 3,i , s
2, f , alf 4,i , s
i
(39)
i
w
i
w
i
w
∑∑ x
+ open f ,2, s = 1
∀ f,s
∑y
2, f , fall ,i , w, s
≤ x2, f , fall ,i , s
∀ f,fall,i,s
∑y
2, f , alf 2,i , w , s
= x2, f ,alf 2,i , s
∀ f,i,s
∑y
2, f , alf 3,i , w , s
= x2, f , alf 3,i , s
∀ f,i,s
∑y
2, f , alf 4,i , w , s
= x2, f ,alf 4,i , s
∀ f,i,s
2, f , c ,i , s
c
(40)
∀s
∑x
i
(38)
≤ PotatoContract
i
i
(37)
∀ f,s
i
w
(41)
w
(42)
w
(43)
w
(44)
∑∑ y
i
+ ∑∑ y1, f , gcorn ,i , w, s + ∑∑ y1, f , scorn ,i , w, s
2, f , gcorn ,i , w , s
w
i
w
i
w
+ H 6 f , gcorn + H 6 f , scorn ≤ 2
(45)
∑∑ y
i
(46)
i
(47)
2, f ,barley ,i , w , s
w
i
i
w
+ H 6 f , gcorn + H 6 f , scorn ≤ 2
w
2, f ,c ,i , w , s
w
∀ f,s
w
+ ∑∑ y1, f ,wheat ,i ,w,s ≤ 1
i
i
+ ∑∑ y1, f ,barley ,i ,w,s
i
∑∑∑ y
c
+ ∑∑ y1, f , gcorn ,i , w, s + ∑∑ y1, f , scorn ,i , w, s
2, f , scorn ,i , w , s
w
∑∑ y
∀ f,s
∀ f,s
w
=1
∀ f,s
68
where,
t = a crop year within the 6-year planning horizon, with possible values of 1
through 6, or within the 6-year period following the planning horizon, with
possible values of 7 through 12.
f = the field in which the cropping activity takes place {F1,…, F10}.
c = the crop {onion, potato, sugar beet, wheat, barley, grain corn, silage corn,
alfalfa (1st through 4th year), fallow, and open}
i = the irrigation technology {furrow, reuse furrow, solid set, wheel line, center
pivot, drip}
w = the deficit irrigation level {0.0, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0}
ρs = probability of the 6-year water supply scenario s
r = interest rate on operating loans and savings
d = discount rate
FAC = number of acres per field (fields assumed to be equal size)
pc = price received per unit of crop c
yldc,i,w = yield per acre of crop c, under irrigation technology i, and deficit
irrigation level w
jc,i = cost of spring planting per acre of crop c, under irrigation technology i.
hc = cost of fall preparation or planting per acre of crop c
fxdcost = fixed cost per acre of land owned, such as a water district fee per acre
and land taxes
maxyldc,i = maximum yield for crop c, under irrigation technology i, given no
water deficit
kyc = yield response coefficient for crop c, which reflects sensitivity to water stress
ETmaxc = gross water requirement of crop c over the growing season to achieve
maximum yield
Ppt = precipitation received during the growing season, which reduces irrigation
requirements
69
IrrigEffic = the proportion of water delivered to the field that reaches the crop root
zone
Water = per acre water allotment for the growing season
TotAcres = total number of acres available for cropping activities
RRateonion = rental rate of an acre eligible for onions (i.e. an acre not planted to
onions in previous 5 years)
RRateother = rental rate of an acre not eligible for onions
EligOniont,s = acres eligible to be rented for onions in period t of scenario s
EligAlft,i,s = acres of alfalfa with productive lifespan remaining in years 7 through
9 for scenario s; acres inherit the irrigation technology used in year 6
EligOthert,s = acres eligible to be rented for crops other than onions in period t of
scenario s; a function of EligOniont,s and EligAlft,i,s.
NetRvalf,i,d1 = net revenue from alfalfa under irrigation technology i, assuming no
deficit irrigation (w =d1)
H1f,c = the crop c to which field f was planted six years prior to the first year of the
planning horizon (i.e. planted in the first year of the previous (historical)
planning horizon) (=0 if not planted, or 1 if planted)
H2f,c = the crop c to which field f was planted five years prior to the first year of
the planning horizon (=0 if not planted, or 1 if planted)
H3f,c = the crop c to which field f was planted four years prior to the first year of
the planning horizon (=0 if not planted, or 1 if planted)
H4f,c = the crop c to which field f was planted three years prior to the first year of
the planning horizon (=0 if not planted, or 1 if planted)
H5f,c = the crop c to which field f was planted two years prior to the first year of
the planning horizon (=0 if not planted, or 1 if planted)
H6f,c = the crop c to which field f was planted one year prior to the first year of the
planning horizon (=0 if not planted, or 1 if planted)
70
PotatoContract = a fixed acreage of potatoes (expressed as number of fields)
contracted in advance with local processors
openf,t = field f is left unprepared and unplanted in the fall of year t (=0 if not left
open, or 1 if left open). This contrasts to “fallow,” which indicates that a
field is either abandoned (if prepared or planted in the previous fall), or not
planted in the spring (if left open in the previous fall).
The producer’s objective (equation 6) is to maximize the expected
discounted stream of profit from the 6-year planning horizon through the selection
of fall and spring crop activities (x and y, respectively). Decisions made in “crop
year” t consist of fall decisions (xt,f,c,i,s) and spring decisions (yt,f,c,i,w,s). Crops that
are either fall-planted or require fall bed-preparation require the following fall
decisions: 1) number of fields to plant or prepare, and 2) an associated irrigation
technology, i, for each field. Spring decisions for each crop, c, include the
following: 1) number of fields to keep (if c is a fall-planted crop) or number of
fields to plant (if c is spring-planted), 2) an irrigation technology, i, for each field
(note: for some crops, decisions made in the preceding fall impose an irrigation
technology on the spring decision), and 3) a deficit irrigation level, w, for each
field. The optimal choice of x and y depends on past, current, and expected future
water supplies, denoted by s.
Economic profit for a particular crop year of the planning horizon, given
water supply scenario, s, is described in (7). Crop mix, output price, number of
acres planted, yield per acre, and cost of spring and fall activities partly determine
profit. Fixed costs (which include land taxes and a water district charge), and the
opportunity cost of money and time also influence profit. A 7% interest rate (r) is
charged for short-term operating loans (Stanger 2005, personal communication).
The opportunity cost of investing equity funds in the farm is also assumed to be
7%. It has been argued that the rate charged for equity funds should be less than
71
the rate charged for borrowed funds, because the commercial lending rate includes
fees that are not relevant to equity funds (American Agricultural Economics
Association Task Force 1998, p33). However, it is not possible to accommodate a
separate interest rate for each source of funds in this model. Time preferences are
captured with a 5% discount rate (d). This rate strikes a balance between a
conservative discount rate of 3% (the average real return on a risk-free asset
(American Agricultural Economics Association Task Force 1998, p33)), and a
higher discount rate (7%) that is based on the assumed interest rate. The effect of
choosing a lower versus higher discount rate on the model’s solution is discussed
in chapter 5.
The constant, FAC, which represents the size of each field and appears
first in the profit calculation, is necessary when x and y are binary variables. For
example,
∑y
1, f ,c ,i , w , s1
calculates the number of fields planted to crop c under
f
irrigation technology i and deficit irrigation level w in the spring of year 1 for
water scenario s1. This integer has to be multiplied by the acres per field (FAC)
before profit is calculated, because the revenue and cost data are per acre, not per
field.
Yield for crop c, under irrigation technology i and deficit irrigation level
w, is calculated in equation 8, which is a linear yield response function popularized
by Doorenbos and Kassam (1979). Water is assumed to be the only limiting input
to crop yield. The degree to which actual crop yield (yldc,i,w) deviates from
maximum yield (maxyldc,i) in a particular year is a function of the crop’s
sensitivity to water stress (indicated by the empirically-based coefficient kyc),
precipitation received during the growing season (Ppt), and the proportion (w) of
the crop’s maximum irrigation water requirement (ETmaxc-Ppt) actually provided.
This formulation of the yield response function assumes water deficits occur at an
equal proportion across the entire growing season. It is preferable to model
strategic deficit irrigation, in which crops are deficit irrigated during their least-
72
sensitive growth stages. Data are insufficient, unfortunately, to model this
approach. The season-long deficit approach likely overestimates yield losses
associated with a particular deficit level. Thus the model is likely to choose deficit
irrigation as an optimal strategy less frequently than a model that assumes strategic
deficit irrigation.
Terminal values are introduced in the model via equation 9. Terminal
values are needed to capture the following two sources of future profit: 1) alfalfa
planted or maintained in year 6 that has productive value in years 7 through 9, and
2) the rental value of land in years 7 through 12. Decisions made in years 1
through 6 impact the flow of profit from years 7 through 12; equation 9 is an
attempt to incorporate this dynamic relationship into the decision problem.
Equations (10) through (45) are the detailed representation of the general
constraints presented in section 4.1, specifically equations (2) through (5).
Equation 10 constrains the sum of water use across all fields, accounting for the
application efficiency of various irrigation technologies, to no more than the
farm’s total water allotment. Total water allotted equals the per acre water
allotment (set by the irrigation district) multiplied by total acres owned or leased.
This equation must be met in every year of every water supply scenario.
Equations 11 and 12 prevent crop-irrigation-deficit combinations that are
not observed in the area from entering the solution. Cost and yield data are not
available for combinations not observed in the area, so they are not included in the
model. The GAMS language uses set notation to reduce the volume of required
code. A by-product of this notation, however, is that cropping activities not
currently practiced in the area are created through the set notation. Suppose, for
example, that set c = {onions, corn} and set i = {drip, center pivot}. Set notation
allows the modeler to specify one equation that applies to every (c, i) combination,
rather than specifying one equation for each combination. Suppose, however, that
not all (c, i) combinations occur in the study area; for example, (corn, drip) does
73
not occur in the Vale Oregon Irrigation District. Equations 11 and 12 prevent this
combination from entering the solution. Equation 13 states that cropping activities
can take on binary values only. That is, a cropping activity can either be
implemented in a particular field (i.e. take on the value 1), or not implemented (i.e.
take on the value 0). This is in contrast to a continuous definition of cropping
activities, in which the activity variable could take on any continuous value
representing the number of acres on which the activity is implemented.
Equations 14 through 21 constrain the scope of specific crop activities in
the fall of year 1 to reflect agronomic rules that prevent pests and diseases. These
rules, derived from conversations with producers, represent agronomic guidelines
that they adhere to quite rigidly. It is beyond the scope of this study to test the
economic optimality of these rules. Biological response functions that capture pest
and disease dynamics are not readily available, and are therefore not directly
included in the economic decision model. These functions are captured through
crop rotation constraints instead.
Interpretation of a few equations will help elucidate the nature of the
agronomic constraints. Equation 14 prevents the planting of small grains (wheat
and barley) on the same acreage in two consecutive years. It specifically states
that field f can be planted to wheat in year 1 if it was not planted to wheat or barley
in year 6 of the historic period (i.e. H6). The historic period consists of the six
years that immediately precede the current planning horizon; historic crop
activities are exogenous to the decision model. Equation 14 states, algebraically,
that the sum of the listed activities (each of which can take the value of 0 or 1)
cannot exceed 1. Equations 15 through 17 are the equivalent to (14) for other
crops. Note that sugar beets, onions, and potatoes require four to five years
between plantings to avoid pests and diseases. These agronomic practices create
inter-year dynamics.
74
Equation 18 states, for year 1, that the number of fields allocated to
potatoes cannot exceed the “PotatoContract,” regardless of the water scenario.
Potatoes in the study area are grown exclusively under contract with local
processors, so producers are constrained to the quantity that the processor requests.
A relatively small portion of onions and sugar beets are also grown under contract.
It was decided, however, to exclude this option from the model. Equations 19
through 21 require the producer to maintain alfalfa that is one or more years old
through its fourth year of production. The producer does, however, have an
opportunity to abandon newly planted alfalfa in its first spring. Alfalfa is used in
crop rotations to enhance soil quality; equations 19 through 21 ensure that alfalfa
is left in place sufficiently long to accomplish this. Equation 22 forces the
producer to make a fall decision for each field in year 1; they can choose to
prepare, plant, or leave each field open.
Equations 23 through 29 constrain spring crop activities in year 1.
Equation 23 limits the spring acreage of each fall-planted or prepared crop to no
more than the number of fields planted or prepared in the preceding fall. Winter
wheat acreage, for example, is planted exclusively in the fall; therefore, wheat
acreage cannot be increased in the spring. Onion acreage, which is prepared in the
fall, cannot typically be increased in the spring due to adverse field conditions.
Equation 23 therefore generates intra-year dynamics. Equations 24 through 26
simply transfer fall alfalfa acreage to spring alfalfa acreage, thus preventing the
abandonment of alfalfa stands that are one or more years old.
Equation 27 states that corn (grain or silage) cannot be planted in the same
field more than two consecutive years. Algebraically, field f can be planted to
grain corn in year 1 if it was planted to grain or silage corn in year H6 but not year
H5, or if it was planted to corn in year H5 but not year H6. Equation 28 presents
the same constraint for silage corn. Equation 30 must accompany equations 27
and 28 for them to perform correctly. It states that each field can be planted in the
75
spring to only one crop-irrigation-deficit combination; fallowing is included in the
list of spring crops. Equations 27 and 28 each sum over several corn-irrigationdeficit combinations for year 1, and the sums are allowed to equal 2; thus, without
equation 30, one field could be planted to two different combinations in the same
year. Equation 29 expresses the agronomic constraint for spring-planted barley.
The equations explained above are defined for year 1 only. Equations 31 through
47 essentially repeat this block for year 2. Blocks for years 3, 4, 5, and 6 are
similar in content, and therefore not presented here.
Care must be taken in constructing the above constraints, due to the
stochastic water supply. First, all constraints must be met in every water supply
scenario. The water constraint in equation 10, for example, must be met in the
event of a full or dry spring. Additionally, constraints must be constructed to
properly account for past water supply conditions. The number of fields planted to
onions in year 4 of water scenario [Full Dry Full Full ___ ___ ], for example,
cannot exceed the number of fields that remain eligible for onions, which is
determined by cropping activities during the three preceding years, i.e. activities in
scenario [Full Dry Full ___ ___ ___ ]. Use of the subscripts s1 through s6 ensures
that current activities are constrained by their respective water supply histories.
4.4
Details of the Continuous Variables Model
Equations of the continuous variables model are presented next, followed
by their interpretation. Although the model is similar in general structure to the
binary variables model, variables differ slightly in their subscripts and definition.
⎡ 6 ⎛ 1
⎤
⎞
π t , s ( xt , s , yt , s ) ⎟
⎢∑ ⎜
⎥
t
t =1 ⎝ (1 + d )
⎠
⎢
⎥
(6) Max Es Π ( x, y; s ) = ∑ ρ s ⋅
⎢
12
x, y
⎛ 1
⎞⎥
s
⎢+∑ ⎜
⎥
(
,...
)
y
y
π
t ,s
t − 6, s
6, s ⎟
t
⎠ ⎦⎥
⎣⎢ t =7 ⎝ (1 + d )
where for t =1,...,6
76
π t ,s ( xt ,s , yt , s ) = ∑∑∑ ( pc yt ,c ,i ,w,s yldc ,i ,w − jc ,i yt ,c,i ,w,s − hc,i xt ,c ,i ,s )
(7)
c
i
w
⎛ ∑∑∑ ( jc,i yt ,c,i , w, s + hc,i xt ,c ,i , s ) ⎞
⎟
− fxd cos t − r ⋅ ⎜ c i w
⎜ + fxd cos t − π ( x , y ) ⎟
t −1, s
t −1, s
t −1, s
⎝
⎠
and
(8)
(9)
⎡ ⎛
⎛ w ⋅ ( ET max c − Ppt ) + Ppt ⎞ ⎞ ⎤
yldc ,i , w = max yldc ,i ⋅ ⎢1 − ⎜⎜ kyc ⋅ ⎜1 −
⎟ ⎟⎟ ⎥
ET max c
⎢⎣ ⎝
⎝
⎠ ⎠ ⎥⎦
where for t =7,...,12
π t ,s ( yt −6, s ,... y6,s ) = RRateonion ⋅ EligOniont , s ( yt −6,s ,... y6, s )
+ RRateother ⋅ EligOthert , s ( yt −6, s ,... y6, s )
⎛
⎞
+ ⎜ ∑ NetRvalf ,i ,d 1 ⋅ EligAlft ,i , s ( y6, s ) ⎟ − fxd cos t
⎝ i
⎠
⎛
⎞
−r ⋅ ⎜ ∑ ( jalf ,i ⋅ EligAlft ,i , s ( y6, s ) ) + fxd cos t − π t −1, s ( yt −6, s ,... y6, s ) ⎟
⎝ i
⎠
subject to
(10)
⎛ ET max c − Ppt ⎞
⎟ ⋅ yt ,c,i , w, s ≤ Watert , s ⋅ TotAcres
⎝ IrrEffici
⎠
∑∑∑ w ⋅ ⎜
c
i
w
∀ t, s
(11)
xt ,c ,i , s = 0 for some t,c,i,s
(12)
y t,c,i,w,s = 0
(13)
xt ,c ,i , s , yt ,c ,i , w, s ≥ 0
∀ t,c,i,w,s
(14)
∑x
≤ TotAcres −H 6wheat − H 6barley
∀s
∑x
≤ TotAcres − H 6sugbt − H 5sugbt − H 4sugbt − H 3sugbt
∑x
≤ TotAcres −H 6onion − H 5onion − H 4onion
for some t,c,i,w,s
1, wheat ,i , s
i
(15)
1, sugbt ,i , s
∀s
i
(16)
1,onion ,i , s
i
(17)
∑x
− H 3onion − H 2onion
1, potato ,i , s
∀s
≤ TotAcres −H 6 potato − H 5 potato − H 4 potato
i
− H 3 potato − H 2 potato
(18)
∑x
1, potato ,i , s
≤ PotatoContract
∀s
∀s
i
(19)
∑x
1, alf 2,i , s
i
= H 6alf 1
∀ i,s
77
(20)
∑x
= H 6alf 2
∀ i,s
∑x
= H 6alf 3
∀ i,s
1, alf 3,i , s
i
(21)
1, alf 4,i , s
i
(22)
∑∑ x
+ open1 = TotAcres
∀s
∑y
≤ x1, fall ,i , s
∀ fall,i,s
∑y
= x1,alf 2,i , s
∀ i,s
∑y
= x1,alf 3,i , s
∀ i,s
∑y
= x1,alf 4,i , s
∀ i,s
1, c ,i , s
c
(23)
i
1, fall ,i , w , s
w
(24)
1, alf 2,i , w , s
w
(25)
1, alf 3,i , w , s
w
(26)
1, alf 4,i , w , s
w
(27)
∑∑ y
1, gcorn ,i , w, s
i
w
+ ∑∑ y1, scorn ,i , w, s + H 6 gcorn + H 6scorn
i
w
+ H 5gcorn + H 5scorn ≤ 2* TotAcres
(28)
∑∑ y
1,barley ,i , w, s
i
w
+ ∑∑ y1, wheat ,i , w, s ≤ TotAcres
i
w
− H 6barley − H 6wheat
(29)
∑∑∑ y
1, c ,i , w , s
c
(30)
i
∑x
w
(31)
∀s
= TotAcres
∀s
≤ TotAcres −∑ ∑ y1, wheat ,i , w, s −∑∑ y1,barley ,i , w, s
2, wheat ,i , s
i
∑x
i
w
i
w
i
− H 5sugbt − H 4 sugbt
∑x
i
w
− H 4onion − H 3onion
∑x
2, potato ,i , s
i
∑x
i
2, potato ,i , s
i
(35)
w
≤ PotatoContract
∀s
∀s
∑x
=∑∑ y1,alf 1,i , w, s
∀s
∑x
=∑∑ y1,alf 2,i , w, s
∀s
2, alf 2,i , s
i
(36)
∀s
≤ TotAcres −∑∑ y1, potato ,i , w, s −H 6 potato
− H 5 potato − H 4 potato − H 3 potato
(34)
∀s
≤ TotAcres −∑∑ y1,onion ,i , w, s −H 6onion − H 5onion
2,onion ,i , s
i
(33)
∀s
w
≤ TotAcres −∑∑ y1, sugbt ,i , w, s − H 6sugbt
2, sugbt ,i , s
i
(32)
∀s
2, alf 3,i , s
i
i
w
i
w
78
(37)
∑x
2, alf 4,i , s
i
(38)
i
∀s
w
∑∑ x
+ open2, s = TotAcres
∀s
∑y
2, fall ,i , w , s
≤ x2, fall ,i , s
∀ fall,i,s
∑y
2, alf 2,i , w, s
= x2,alf 2,i , s
∀ i,s
∑y
2, alf 3,i , w , s
= x2, alf 3,i , s
∀ i,s
∑y
2, alf 4,i , w, s
= x2,alf 4,i , s
∀ i,s
2, c ,i , s
c
(39)
=∑∑ y1,alf 3,i , w, s
i
w
(40)
w
(41)
w
(42)
w
(43)
∑∑ y
i
2, gcorn ,i , w , s
w
+ ∑∑ y2, scorn,i , w, s + ∑∑ y1, gcorn ,i , w, s
i
w
i
w
+ ∑∑ y1, scorn ,i , w, s + H 6 gcorn + H 6 scorn
i
w
≤ 2* TotAcres
(44)
∑∑ y
i
(45)
2,barley ,i , w , s
w
i
+ ∑∑ y2, wheat ,i , w, s ≤ TotAcres
i
w
−∑∑ y1,barley ,i , w, s + ∑∑ y1, wheat ,i , w, s
∑∑∑ y
c
∀s
i
2, c ,i , w , s
w
i
∀s
w
= TotAcres
∀s
w
where,
t = a crop year within the 6-year planning horizon, with possible values of 1
through 6, or within the 6-year period following the planning horizon, with
possible values of 7 through 12.
c = the crop {onion, potato, sugar beet, wheat, barley, grain corn, silage corn,
alfalfa (1st through 4th year), fallow}
i = the irrigation technology {furrow, reuse furrow, solid set, wheeline, center
pivot, drip}
w = the deficit irrigation level {0.0, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0}
ρs = probability of the 6-year water supply scenario s
r = interest rate on operating loans and savings
79
d = discount rate
pc = price received per unit of crop c
yldc,i,w = yield per acre of crop c, under irrigation technology i, and deficit
irrigation level w
jc,i = cost of spring planting per acre of crop c, under irrigation technology i.
hc = cost of fall preparation or planting per acre of crop c
fxdcost = fixed cost per acre of land owned, such as a water district fee per acre
and land taxes
maxyldc,i = maximum yield for crop c, under irrigation technology i, given no
water deficit
kyc = yield response coefficient for crop c, which reflects sensitivity to water stress
ETmaxc = gross water requirement of crop c over the growing season to achieve
maximum yield
Ppt = precipitation received during the growing season, which reduces irrigation
requirements
IrrigEffic = the proportion of water delivered to the field that reaches the crop root
zone
Water = per acre water allotment for the growing season
TotAcres = total number of acres available for cropping activities
RRateonion = rental rate of an acre eligible for onions (i.e. an acre not planted to
onions in previous 5 years)
RRateother = rental rate of an acre not eligible for onions
EligOniont,s = acres eligible for onions in period t of scenario s
EligAlft,i,s = acres of alfalfa with productive lifespan remaining in years 7 through
9 for scenario s; acres inherit the irrigation technology used in year 6, for
simplicity
EligOthert,s = acres eligible for crops other than onions in period t of scenario s; a
function of EligOniont,s and EligAlft,i,s
80
NetRvalf,i,d1 = net revenue from alfalfa under irrigation technology i, assuming no
deficit irrigation (w =d1)
H1c = acres of crop c planted six years prior to the first year of the planning
horizon (i.e. planted in the first year of the previous (historical) planning
horizon)
H2c = acres of crop c planted five years prior to the first year of the planning
horizon
H3c = acres of crop c planted four years prior to the first year of the planning
horizon
H4c = acres of crop c planted three years prior to the first year of the planning
horizon
H55 = acres of crop c planted two years prior to the first year of the planning
horizon
H6c = acres of crop c planted one year prior to the first year of the planning
horizon
PotatoContract = a fixed acreage of potatoes contracted in advance with local
processors
opent = acres to leave unprepared and unplanted in the fall of year t
Interpretation of the continuous variables model’s equations is similar to
that of the binary variables model. One difference is the lack of subscript f on the
x and y decision variables, which changes x and y from binary variables to
continuous variables, measured in acres rather than fields. Another difference is
the manner in which agronomic constraints are specified. Interpretation of a few
equations will illustrate how they differ from the binary model. Equation 14
prevents the planting of wheat on acreage that was planted to wheat or barley in
the previous year. The constraint specifically defines acres eligible for wheat this
year as total farm acreage less acres planted to wheat or barley last year. This
form of the constraint cannot guarantee that a particular field will not be planted to
81
small grains in two consecutive years, because the continuous model does not
track crop history on a field-by-field basis. However, it does guarantee at the
farm-level that acres allocated to wheat do not exceed eligible acres. The binary
variables model avoids this spatial ambiguity by tracking each field’s crop history.
Equation 27 attempts to capture the agronomic constraint for grain and
silage corn, which stipulates that acreage should not be planted to corn more than
two consecutive years. It is difficult to represent this constraint, however, without
tracking the crop history of individual fields; equation 27 therefore only
approximates it. Equation 27 states that over a three-year period the sum of
acreage planted to grain corn or silage corn cannot exceed twice the total acreage.
The constraint works well if it is assumed that the producer avoids planting corn
on the same acreage in two consecutive years. The illustrations below clarify the
constraint. Suppose in year 1 and 2 that all 350 acres are planted to corn. Equation
27 states that acres planted to corn in years 1, 2, and 3 cannot exceed two times the
total farm acreage (350 acres), which implies that 0 acres are eligible for corn in
year 3 for this example. This coincides with the agronomic practice; corn was
planted in two consecutive years on all 350 acres, therefore, zero acres are eligible
in year 3.
Corn =
350 acres
Corn =
350 acres
Corn ≤ 700
– 350 – 350
≤ 0 acres
Year 1
Year 2
Year 3
Consider a second example in which the constraint also functions well.
Corn =
150 acres
Corn =
350 acres
Corn ≤ 700
– 350 – 150
≤ 200 acres
Year 1
Year 2
Year 3
82
Corn was planted for two consecutive years (years 1 and 2) on 150 acres.
However, 200 acres were planted to corn only in year 2, thus 200 acres are eligible
for corn in year 3. Consider a final example, in which the constraint does not
capture the agronomic practice exactly.
Corn =
200 acres
Corn =
200 acres
Corn ≤ 700
– 200 – 200
≤ 300 acres
Year 1
Year 2
Year 3
It is not clear in the third example whether corn in year 2 was planted on
the same acres as corn in year 1. The overlap is, at a minimum, 50 acres, but could
be as large as 200 acres. Equation 27 assumes that the minimum possible overlap
(50 acres) occurred in year 2, such that 300 acres are eligible for corn in year 3. In
reality, the producer may have overlapped the entire 200 acres, which leaves only
150 acres eligible in year 3. This example highlights the limitations of using a
continuous variables model to represent agronomic practices that operate on
discrete fields. The binary variables model, in contrast, represents the agronomic
practice precisely. The solutions to the continuous and binary models share
similar characteristics, however, including the relative acreage of corn versus other
crops. This suggests that the continuous variables version of the corn constraint
does not lead to serious errors in the solution. The remaining equations are
sufficiently similar in interpretation to those in the binary model, so they will not
be discussed further.
4.5
Alternative Versions of the Binary Variables Model
The binary variables model is used in section 5.1 to establish the “base
case” solution. In sections 5.1 through 5.8, the binary variables model is modified
to conduct sensitivity analyses and facilitate interpretation of the base case
solution. Section 5.2 focuses on sensitivity to the discount and interest rates.
Section 5.3 treats the water supply as certain to explore differences between
83
optimal drought preparedness and response under certainty and uncertainty.
Section 5.4 discusses the importance of inter-year crop dynamics. Section 5.5
introduces crop history by setting parameters H1 through H6 to non-zero values.
Section 5.6 introduces price uncertainty in addition to water supply uncertainty.
Section 5.7 introduces a crop insurance product known as prevented planting
provisions. Section 5.8 resumes water supply uncertainty, but examines the impact
of increased drought frequency and severity. Lastly, the continuous variables
model’s solution is presented in section 5.9, and compared with the base case
solution. For most sections, only small modifications of the base case model are
made (e.g. parameter values are changed). It suffices in these cases to explain the
modification briefly at the beginning of the respective section in chapter 5.
Sections 5.3, 5.6, and 5.7, in contrast, require changes in the model structure that
are sufficient to require more thorough descriptions. These are presented next.
4.5.1 Water Supply Certainty
A deterministic version of the base case model is constructed to tease out
uncertainty’s role in optimal preparedness and response. The deterministic model
assumes that the water supplies for all six years of the planning horizon are known
in the fall of year 1. A specific water supply scenario is therefore assigned a priori
(e.g. 24 acre-inches per acre in each of the six years), and the model chooses fall
and spring activities for years 1 through 6 to maximize the stream of discounted
profit. This model could be used to conduct a passive programming analysis of the
effects of water supply on production (see section 2.2.3). The general model is as
follows:
84
(1)
Max Π ( x, y )
x, y
s.t.
(2)
Ax = b
(3)
Dy = e
(4)
(5)
Mx + Ny = g
x, y ≥ 0
where all notation referring to states of nature is removed, and some definitions
simplify as follows:
x = Vector containing fall crop decision variables for each year of the planning
horizon.
Example element: x3,f,c,i, which indicates for the fall of year 3, that field f is
prepared for or planted to crop c, under irrigation technology i. Each
element of x is a binary variable, taking on a value of 0 or 1.
y = Vector of spring crop decision variables for each year of the planning horizon.
Example element: y3,f,c,i,w, which indicates for the spring of year 3, that field f
is planted to crop c in the spring of year 3, under irrigation technology i,
and deficit irrigation category w. Each element of y is a binary variable,
taking on a value of 0 or 1.
Π ( x, y ) = The discounted stream of profit that optimal activities x and y generate
over the 6-year period in which they occur, assuming that the water supply
for each of the six years is known a priori.
4.5.2 Price Uncertainty
The base case model must be modified significantly to accommodate
uncertainty of prices. Price uncertainty is represented by three price categories
(Appendix B.3), and is resolved after both fall and spring decisions are made. A
third stage is therefore added to the model, during which the producer learns which
price will be received. The producer has no recourse after the price is revealed.
85
All crops that are grown are assumed to be sold at the market price. Marketing
strategies were beyond the scope of this study. The general price uncertainty
model is presented next, followed by details of the model.
4.5.2.1 THE GENERAL PRICE UNCERTAINTY MODEL
(1)
Max Es , ss Π ( x, y, z; s, ss )
x, y
s.t.
(2)
Ax = b
(3)
Dy = e
(4)
(5)
Mx + Ny + Rz = g
x, y , z ≥ 0
where
s = A random vector that represents water supplies over a 3-year planning horizon.
Each realization of s consists of 3 components (s1 s2 s3), which indicate the
state of nature (water supply category) revealed in each of the three years.
That is, s1 represents the state of nature revealed in year 1, s2 the state of
nature revealed in year 2, etc.
ss = A random vector that represents the price of onions in each of the 3 years of
the planning horizon.
Assuming 2 possible states of the water supply (Dry or Full) and 3 possible
states of the onion price (Lo, Med, Hi) in each of 3 years, 216 three-year
scenarios are possible. Scenarios range from [Dry Lo Dry Lo Dry Lo] to
[Full Hi Full Hi Full Hi], and every combination between. The scenario
[Dry Lo Full Med Full Hi] indicates (from left to right) that the water
supply in year 1 was Dry, the onion price in year 1 was Lo, the water
supply in year 2 was Dry, the onion price in year 2 was Med, the water
supply in year 3 was Full, and the onion price in year 3 was Hi. Each state
of nature has a probability of occurrence within any given year, denoted
pr(Dry) or pr(Full), and prp(Lo), prp(Med), or prp(Hi). The state of nature
86
in any one year is assumed independent of the state of nature in any other
year. Therefore, the joint probability of a particular three-year scenario is
the product of the probabilities of the states of nature that occur each year.
For example, pr([Dry Lo Full Med Full Hi]) = pr(Dry)*prp(Lo)*pr(Full)*
prp(Med)*pr(Full)*prp(Hi). Historical water allotment and onion price
data and Gaussian quadrature analysis (Featherstone, Baker, and Preckel
1993; Miller and Rice 1983; Preckel and Devuyst 1992) were used to
assign a value and a probability to each state of nature (Appendix B).
x = Vector containing fall crop decision variables for each year of the planning
horizon.
Example element: x3,f,c,i,s1,ss1,s2, which indicates for the fall of year 3, that field
f is prepared for or planted to crop c, under irrigation technology i, given
the states of nature revealed in past years 1 and 2. Each element of x is a
binary variable, taking on a value of 0 (if the crop/irrigation combination
(c,i) is not chosen for field f) or 1 (if the crop/irrigation combination (c,i) is
chosen for field f).
y = Vector of spring crop decision variables for each year of the planning horizon.
Example element: y3,f,c,i,w,s1,ss1,s2,ss2,s3, which indicates for the spring of year 3,
that field f is planted to crop c in the spring of year 3, under irrigation
technology i, and deficit irrigation category w, given the states of nature
revealed in past years 1, 2, and the present year 3. Each element of y is a
binary variable, taking on a value of 0 (if the crop-irrigation-deficit
combination (c,i,w) is not chosen for field f) or 1 (if the crop-irrigationdeficit combination (c,i,w) is chosen for field f).
z = Vector of variables that represents the crops sold at the market price for each
year of the planning horizon.
Example element: z3,f,c,i,w,s1,ss1,s2,ss2,s3,ss3, which indicates for the post-harvest of
year 3, that field f, which is planted to crop c in the spring of year 3, under
87
irrigation technology i, and deficit irrigation category w, given the states of
nature revealed in past years 1, 2, and the present year 3, is sold at market
price ss. Each element of z is a binary variable, taking on a value of 0 (if
the crop-irrigation-deficit combination (c,i,w) on field f is not sold) or 1 (if
the crop-irrigation-deficit combination (c,i,w) on field f is sold).
Π ( x, y, z; s, ss ) = Vector containing the profit outcome for each scenario. An
individual element of the vector is the discounted stream of profit that
optimal activities x, y, and z generate over the 3-year period in which they
occur, for a particular scenario. Terminal land rental values are also
included, as a function of activities in the 3-year period. A terminal value
for alfalfa acreage that remains in production after year 3 is also included.
A, D = Matrices of coefficients that describe fall and spring activities’ resource
use.
b, e = Vectors of resource availability, such as land and water, which vary by state
of nature for some resources.
M, N = Matrices of coefficients that relate activities in different time periods to
each other (intra- and inter-year constraints).
g = Vector of parameters that, with M and N above, define relationships between
activities in different time periods.
The above binary discrete stochastic sequential programming model
maximizes the expected stream of profit over a 3-year planning horizon. The
planning horizon is shortened from six to three years because the programming
software, GAMS, can only accommodate ten subscripts on decision variables. A
six-year horizon requires sixteen subscripts. The expectation is taken over the
joint probability of water supply, s, and onion price, ss, which are assumed to have
88
independent discrete probability distributions over a small number of pre-defined
categories (e.g. dry and full; lo, med, and hi ).
Choice variables are contained in the vectors x, y and z. Vector x includes
fall cropping activities, which are chosen under an uncertain future water supply
and onion price. Vector y includes spring cropping activities, which are chosen
after water supply is revealed, but before onion price is revealed. Vector z
includes post-harvest sales activities, which take place after the price is revealed.
It is assumed, however, that all crops grown are sold. The producer therefore has
no recourse after price is revealed; this is in contrast to the recourse options
available after the water supply is revealed (e.g. fallowing, spring-planted crops,
deficit irrigation). Fall, spring, and post-harvest activities are chosen for each
year of the three-year planning horizon, for each water supply scenario, (e.g. [Full
Lo Full Hi Dry Med]). Fall and spring activities are constrained by resource
availability, as expressed in equations (2) and (3). Equation (4) describes dynamic
interactions in the cropping system, including how fall activities restrict spring
activities, and how spring activities restrict post-harvest activities (intra-year
dynamics), and how activities in year t restrict activities in subsequent years (interyear dynamics). The timing of decisions relative to the availability of water
supply information is presented in figure 4.2. Past water supplies and onion prices
are known, but future water supplies and onion prices are uncertain.
89
Year 1
p(hi)=0.25
p(med)=0.50
p(lo)=0.25
full
y1|x1
x1
dry
water &
price
uncertain
(fall)
y1|x1
water
price
revealed uncertain
(spring) (spring)
hi
z1|y1
med
z1|y1
lo
z1|y1
hi
z1|y1
med
z1|y1
lo
z1|y1
price
revealed
(postharvest)
x2|z1
profit outcome
p(wet)=0.6
p(dry)=0.4
Year 2
x2|z1
water &
price
uncertain
(fall)
full
y2|x2
dry
y2|x2
full
y2|x2
dry
y|x 2
water
revealed
(spring)
Figure 4.2. Decision tree representation of decision-making under water and price uncertainty. Fall cropping activities (xt)
are chosen given an uncertain spring water supply and post-harvest onion price. The spring water supply is then revealed
(full or dry), after which spring cropping activities (yt) are chosen under uncertain onion price. The onion price is then
revealed (hi, med, or lo), after which post-harvest sales activities (zt) are chosen. This sequence continues for all three
years of the planning horizon.
89
90
4.5.2.2 DETAILS OF THE PRICE UNCERTAINTY MODEL
⎡ 3 ⎛ 1
⎞⎤
π t , s , ss ( xt , s , ss , yt , s , ss , zt , s , ss ) ⎟ ⎥
⎢∑ ⎜
t
t =1 ⎝ (1 + d )
⎠⎥
(6) Max Es , ss Π ( x, y; s, ss ) = ∑∑ ρ s ρ ss ⋅ ⎢
⎢ 9 ⎛ 1
x, y
⎞⎥
s ss
⎢+∑ ⎜
⎥
π
y
y
(
,...
)
,
,
3,
,
3,
,
−
t
s
ss
t
s
ss
s
ss
⎟
t
⎢⎣ t = 4 ⎝ (1 + d )
⎠ ⎥⎦
where for t =1, 2, 3
⎛ ∑ FAC ⋅ ( pc , ss yld c ,i , w zt , f ,c ,i , w, s , ss ⎞
⎟
(7) π t , s , ss ( xt , s , ss , yt , s , ss , zt , s , ss ) = ∑∑∑ ⎜ f
⎟
c
i
w ⎜− j y
⎝ c ,i t , f ,c ,i , w, s , ss − hc ,i xt , f ,c ,i , s , ss ) ⎠
⎛
⎛
⎞⎞
⎜ ∑∑∑ ⎜ ∑ FAC ⋅ ( jc ,i yt , f ,c ,i , w, s , ss + hc ,i xt , f ,c ,i , s , ss ) ⎟ ⎟
− fxd cos t − r ⋅ ⎜ c i w ⎝ f
⎠⎟
⎜ + fxd cos t − π
⎟
t −1, s , ss ( xt −1, s , ss , yt −1, s , ss , zt −1, s , ss )
⎝
⎠
and
(8)
⎡ ⎛
⎛ w ⋅ ( ET max c − Ppt ) + Ppt ⎞ ⎞ ⎤
yld c ,i , w = max yld c ,i ⋅ ⎢1 − ⎜⎜ kyc ⋅ ⎜1 −
⎟ ⎟⎟ ⎥
ET max c
⎢⎣ ⎝
⎝
⎠ ⎠ ⎥⎦
where for t =4,...,9
(9) π t , s , ss ( yt −3,s , ss ,... y3, s , ss ) = RRateonion ⋅ EligOniont , s ( yt −3, s , ss ,... y3, s , ss )
+ RRateother ⋅ EligOthert , s , ss ( yt −6, s , ss ,... y6, s , ss )
⎛
⎞
+ ⎜ ∑ NetRvalf ,i , d 1 ⋅ EligAlft ,i , s , ss ( y3, s , ss ) ⎟ − fxd cos t
⎝ i
⎠
⎛ ∑ ( jalf ,i ⋅ EligAlft ,i , s , ss ( y3, s , ss ) ) + fxd cos t ⎞
⎟
−r ⋅ ⎜ i
⎜ −π
⎟
⎝ t −1, s , ss ( yt −3, s , ss ,... y3, s , ss )
⎠
subject to
(10)
⎛ ET max c − Ppt ⎞
⎟ ⋅ FAC ⋅ yt , f ,c ,i , w, s , ss
IrrEffici
⎝
⎠
∑ ∑∑∑ w ⋅ ⎜
c
i
w
f
≤ Watert , s ⋅ TotAcres
∀ t, s, ss
(11) xt , f ,c ,i , s , ss = 0
for some t,c,i,s,ss
∀f
(12) y t,f,c,i,w,s,ss = 0
for some t,c,i,w,s,ss
∀f
(13) xt , f ,c ,i , s , ss , yt , f ,c ,i , w, s , ss , zt , f ,c ,i , w, s , ss = 0 or 1 ∀ t,f,c,i,w,s, ss
91
(14)
∑x
+H 6 f , wheat + H 6 f ,barley ≤ 1
∑x
+ H 6 f , sugbt + H 5 f , sugbt + H 4 f , sugbt
1, f , wheat ,i
∀f
i
(15)
1, f , sugbt ,i
i
+ H 3 f , sugbt ≤ 1
(16)
∑x
∀f
+H 6 f ,onion + H 5 f ,onion + H 4 f ,onion
1, f , onion ,i
i
+ H 3 f ,onion + H 2 f ,onion ≤ 1
(17)
∑x
1, f , potato ,i
∀f
+H 6 f , potato + H 5 f , potato + H 4 f , potato
i
+ H 3 f , potato + H 2 f , potato ≤ 1
(18)
∑∑ x
1, f , potato ,i
i
(19)
∀f
≤ PotatoContract
f
∑x
= H 6 f ,alf 1
∀ f,i
∑x
= H 6 f ,alf 2
∀ f,i
∑x
= H 6 f ,alf 3
∀ f,i
1, f , alf 2,i
i
(20)
1, f , alf 3,i
i
(21)
1, f , alf 4,i
i
(22)
∑∑ x
1, f , c ,i
c
(23)
+ open f ,1 = 1
∀f
i
∑y
≤ x1, f , fall ,i
∀ f,fall,i,s
∑y
= x1, f ,alf 2,i
∀ f,i,s
∑y
= x1, f ,alf 3,i
∀ f,i,s
∑y
= x1, f ,alf 4,i
∀ f,i,s
1, f , fall ,i , w , s
w
(24)
1, f , alf 2,i , w , s
w
(25)
1, f , alf 3,i , w , s
w
(26)
1, f , alf 4,i , w , s
w
(27)
∑∑ y
1, f , gcorn ,i , w , s
i
+ H 6 f , gcorn + H 6 f , scorn
w
+ H 5 f , gcorn + H 5 f , scorn ≤ 2
(28)
∑∑ y
1, f , scorn ,i , w , s
i
+ H 6 f , gcorn + H 6 f , scorn
w
+ H 5 f , gcorn + H 5 f , scorn ≤ 2
(29)
(30)
∀ f,s
∑∑ y
1, f ,barley ,i , w , s
i
w
c
i
∑∑∑ y
1, f , c ,i , w , s
w
∀ f,s
+ H 6 f ,barley + H 6 f , wheat ≤ 1
∀ f,s
=1
∀ f,s
92
(31)
z1, f ,c ,i , w, s , ss = y1, f ,c ,i , w, s
(32)
∑x
+ ∑∑ y1, f , wheat ,i , w, s + ∑∑ y1, f ,barley ,i , w, s ≤ 1
∑x
+ ∑∑ y1, f , sugbt ,i , w, s + H 6 f , sugbt
2, f , wheat ,i , s , ss
i
(33)
2, f , sugbt ,i , s , ss
i
∀ f,c,i,w,s,ss
i
w
i
w
i
+ H 5 f , sugbt + H 4 f , sugbt ≤ 1
(34)
∑x
∀ f,s,ss
+ ∑∑ y1, f ,onion ,i , w, s +H 6 f ,onion + H 5 f ,onion
2, f , onion ,i , s , ss
i
i
w
+ H 4 f ,onion + H 3 f ,onion ≤ 1
(35)
∑x
2, f , potato ,i , s , ss
i
∀ f,s,ss
+ ∑∑ y1, f , potato ,i , w, s +H 6 f , potato + H 5 f , potato
i
w
+ H 4 f , potato + H 3 f , potato ≤ 1
(36)
∑∑ x
2, f , potato ,i , s , ss
f
(37)
∀ f,s,ss
∑x
=∑∑ y1, f ,alf 2,i , w, s
∀ f,s,ss
∑x
=∑∑ y1, f ,alf 3,i , w, s
∀ f,s,ss
∑∑ x
+ open f ,2, s , ss = 1
∀ f,s,ss
∑y
2, f , fall ,i , w , s , ss
≤ x2, f , fall ,i , s , ss
∀ f,fall,i,s,ss
∑y
2, f , alf 2,i , w , s , ss
= x2, f ,alf 2,i , s , ss
∀ f,i,s,ss
∑y
2, f , alf 3,i , w, s , ss
= x2, f , alf 3,i , s , ss
∀ f,i,s,ss
∑y
2, f , alf 4,i , w , s , ss
= x2, f ,alf 4,i , s , ss
∀ f,i,s,ss
2, f , alf 3,i , s , ss
2, f , alf 4,i , s , ss
2, f , c ,i , s , ss
c
(41)
∀ s,ss
=∑∑ y1, f ,alf 1,i , w, s
2, f , alf 2,i , s , ss
i
(40)
≤ PotatoContract
∑x
i
(39)
∀ f,s,ss
i
i
(38)
∀ f,s,ss
w
i
w
i
w
i
w
i
w
(42)
w
(43)
w
(44)
w
(45)
∑∑ y
i
2, f , gcorn ,i , w , s , ss
w
+ ∑∑ y1, f , gcorn ,i , w, s + ∑∑ y1, f , scorn ,i , w, s
i
w
i
w
+ H 6 f , gcorn + H 6 f , scorn ≤ 2
(47)
∑∑ y
i
w
2, f , scorn ,i , w , s , ss
∀ f,s,ss
+ ∑∑ y1, f , gcorn ,i , w, s + ∑∑ y1, f , scorn ,i , w, s
i
w
+ H 6 f , gcorn + H 6 f , scorn ≤ 2
i
w
∀ f,s,ss
93
(48)
∑∑ y
i
(49)
i
+ ∑∑ y1, f ,barley ,i , w, s
i
w
+ ∑∑ y1, f , wheat ,i , w, s ≤ 1
∑∑∑ y
c
(50)
2, f ,barley ,i , w, s , ss
w
i
2, f ,c ,i , w, s , ss
∀ f,s,ss
w
=1
∀ f,s,ss
w
z2, f ,c ,i , w, s , ss = y2, f ,c ,i , w, s , ss
∀ f,c,i,w,s,ss
where,
t = a crop year within the 3-year planning horizon, with possible values of 1
through 3, or within the 6-year period following the planning horizon, with
possible values of 4 through 9.
f = the field in which the cropping activity takes place {F1,…, F10}.
c = the crop {onion, potato, sugar beet, wheat, barley, grain corn, silage corn,
alfalfa (1st through 4th year), fallow}
i = the irrigation technology {furrow, reuse furrow, solid set, wheeline, center
pivot, drip}
w = the deficit irrigation level {0.0, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0}
ρs = probability of the 3-year water supply scenario s
ρss = probability of the 3-year price scenario ss
r = interest rate on operating loans and savings
d = discount rate
FAC = number of acres per field (fields assumed to be equal size)
pc,ss = price received per unit of crop c in price scenario ss
yldc,i,w = yield per acre of crop c, under irrigation technology i, and deficit
irrigation level w
jc,i = cost of spring planting per acre of crop c, under irrigation technology i.
hc = cost of fall preparation or planting per acre of crop c
fxdcost = fixed cost per acre of land owned, such as a water district fee per acre
and land taxes
94
maxyldc,i = maximum yield for crop c, under irrigation technology i, given no
water deficit
kyc = yield response coefficient for crop c, which reflects sensitivity to water stress
ETmaxc = gross water requirement of crop c over the growing season to achieve
maximum yield
Ppt = precipitation received during the growing season, which reduces irrigation
requirements
IrrigEffic = the proportion of water delivered to the field that reaches the crop root
zone
Water = per acre water allotment for the growing season
TotAcres = total number of acres available for cropping activities
RRateonion = rental rate of an acre eligible for onions (i.e. an acre not planted to
onions in previous 5 years)
RRateother = rental rate of an acre not eligible for onions
EligOniont,s,ss = acres eligible for onions in period t of scenario s,ss
EligAlft,i,s,ss = acres of alfalfa with productive lifespan remaining in years 4
through 6 for scenario s,ss; acres inherit the irrigation technology used in
year 3, for simplicity
EligOthert,s,ss = acres eligible for crops other than onions in period t of scenario
s,ss; a function of EligOniont,s,ss and EligAlft,i,s,ss
NetRvalf,i,d1 = net revenue from alfalfa under irrigation technology i, assuming no
deficit irrigation (w =d1)
H1f,c = the crop c to which field f was planted six years prior to the first year of the
planning horizon (i.e. planted in the first year of the previous (historical)
planning horizon) (=0 if not planted, or 1 if planted)
H2f,c = the crop c to which field f was planted five years prior to the first year of
the planning horizon (=0 if not planted, or 1 if planted)
95
H3f,c = the crop c to which field f was planted four years prior to the first year of
the planning horizon (=0 if not planted, or 1 if planted)
H4f,c = the crop c to which field f was planted three years prior to the first year of
the planning horizon (=0 if not planted, or 1 if planted)
H5f,c = the crop c to which field f was planted two years prior to the first year of
the planning horizon (=0 if not planted, or 1 if planted)
H6f,c = the crop c to which field f was planted one year prior to the first year of the
planning horizon (=0 if not planted, or 1 if planted)
PotatoContract = a fixed acreage of potatoes (expressed as number of fields)
contracted in advance with local processors
openf,t = leave field f unprepared and unplanted in the fall of year t (=0 if not left
open, or 1 if left open)
The producer’s objective (equation 6) is to maximize the expected
discounted stream of profit from the 3-year planning horizon through the selection
of fall, spring, and post-harvest crop activities (x, y, and z respectively). Decisions
made in “crop year” t consist of fall decisions (xt,f,c,i,s,ss), spring decisions
(yt,f,c,i,w,s,ss), and post-harvest decisions (zt,f,c,i,w,s,ss). Crops that are either fallplanted or require fall bed-preparation require the following fall decisions: 1)
number of fields to plant or prepare, and 2) an associated irrigation technology, i,
for each field. Spring decisions for each crop, c, include the following: 1) number
of fields to keep (if c is a fall-planted crop) or number of fields to plant (if c is
spring-planted), 2) an irrigation technology, i, for each field (note: for some crops,
decisions made in the preceding fall impose an irrigation technology on the spring
decision), and 3) a deficit irrigation level, w, for each field. Post-harvest decisions
are not actually decisions; all spring-grown crops are simply sold at the revealed
market prices (equations 31 and 50 above).
96
All crops’ prices are revealed post-harvest; however, onions are the only
crop to which different prices are assigned for the alternative states of nature. This
implies price certainty for all other crops. The optimal choice of x, y, and z
depends on past, current, and expected future water supplies and onion prices,
denoted by s and ss, respectively. Interpretation of the price uncertainty model is
otherwise similar to that of the binary variables model.
4.5.3 Prevented Planting Provisions
The base case model is modified to accommodate prevented planting
provisions primarily by adding new cropping activities to the existing set. The
new activities reflect the producer’s option to purchase alternative levels of multiperil crop insurance (each with a prevented planting provision) for onions,
potatoes, sugar beets, and wheat. Insurance can be purchased on a field-by-field
basis. Crop insurance policies are purchased in the fall. If the water supply is
revealed dry, the producer then chooses whether to abandon the crop and receive a
prevented planting payment, or to plant the crop. Claims for post-planting
disasters, such as hail, pests, freeze, or abnormally high temperatures are not
modeled. Crop insurance is also not offered in this model for spring-planted crops.
These would require a third stage in each year of the model, in which the
uncertainty about these events would be resolved. Solution of the model is
sufficiently difficult with only two stages per year, so the addition of a third year
remains for future work.
Other modifications are necessary to model prevented planting provisions.
The following parameters, which are used to calculate a prevented planting
payment, are added to the model: approved yield, MPCI coverage level, price
election, PP coverage level, and the premium per acre paid (Appendix C). All
parameters are chosen based on options available in year 2004, so that premiums
are consistent with other costs. Yield and price elections are set to the levels that
most closely match maximum yield and average historical price, as defined in the
97
base case model. These assumptions could be altered, however, to conduct
sensitivity analyses. Fall costs of insured crops are adjusted to include the
premium paid for insurance coverage. Lastly, profit is redefined to reflect
prevented planting payments received if insured crops are abandoned in a dry year.
No payment is received if the crop is abandoned during a full allotment year
(water must be insufficient to receive a payment), or if the insured crop is planted
successfully. The structure of the base case model remains unchanged, so it is not
presented again.
4.6
Model Validation
The model described above abstracts from some aspects of the complex
decision environment in which producers operate. Such abstraction is needed in
order to focus on water supply uncertainty. Isolation of this aspect of the decision
environment enables the model to identify cropping and profit impacts that might
otherwise be obscured. The cost of this abstraction, however, is that the model
cannot be expected to reproduce outcomes observed on an actual farm. While the
model results display similarities to certain responses observed in the area, such as
types of cropping activities and adoption of specific irrigation technologies,
quantitative validation of such a stylized model is difficult and not particularly
relevant to the objectives of the study. It is important, however, in the absence of
quantitative validation, to express clearly the limitations and appropriate
interpretation of the model’s results.
Recognition of aspects of the producer’s decision environment that are
simplified or excluded from the model is critical. These include, but are not
limited to the following: 1) the assumption of a risk-neutral producer, or exclusion
of risk-aversion, 2) the exclusion of many sources of uncertainty, 3) the discrete
treatment of water supply, 4) the over-simplification of complex capital
management issues (e.g. it is assumed that the producer can acquire and dispose of
machinery and equipment each year without transactions costs, and hire and
98
dismiss labor as needed), and 5) the exclusion of the farm household’s
consumption decision. The first point of departure is sufficient to cause
discrepancies in the model’s solution versus observed cropping activities in the
study area.
These assumptions and simplifications should, however, generally lead to
conservative estimates of drought’s true impact. Inclusion of other sources of
uncertainty, for example, would likely reduce the emphasis on drought
preparedness and thus generate crop plans that leave the producer more vulnerable
to drought. Continuous representation of the state of nature would include drought
events more severe than the discrete categories defined in this model, and hence
increase drought’s profit impact. More accurate representation of short-term
capital constraints would reduce the producer’s flexibility, and thus also increase
drought’s profit impact. Accounting for the consumption-smoothing tendencies of
many farm households (Kwon, Orazem, and Otto 2006; Langemeier and Patrick
1990), rather than excluding consumption from the model entirely, would decrease
farm profit further in every scenario. It is unclear, however, whether the profit
impact of drought would be larger or smaller for a risk-averse producer. A riskaverse producer would choose a plan that generates lower expected profit, but less
variable profit. The difference in profit between a drought scenario and a droughtfree scenario would therefore be less. Whether the current model over- or underestimates drought’s impact for a risk-averse producer would therefore depend on
how one defined “profit impact of drought.”
The implication of the above abstractions is that the model used here is
only one step towards a complete understanding of drought management. The
optimal preparedness and response plans presented in chapter 5 are not
prescriptions to be applied directly in the study area. They do, however, help
elucidate the tradeoffs that water supply uncertainty creates in a dynamic farm
system, and in doing so help clarify for producers and extension educators an
99
overwhelmingly complex decision problem. It will also help policymakers and
others concerned with agricultural drought management think more critically about
the meaning of optimal drought preparedness and response.
Despite its abstractions, the model is based very closely on characteristics
of farms in the study area. Specifically, conversations with producers and other
experts in the study area provided information that was used to construct the
model. Examples include the timing of decisions versus water supply information,
common agronomic practices and the underlying reasons for them, common
drought management tools, and insights about the aspects of drought they find
most challenging. These and other details inspired the model’s two-stage
framework and intra- and inter-year dynamics. In addition to producer input,
enterprise budgets constructed for the county in which the study area is located
were the primary source of data for model calibration. In summary, the model, as
with all models, is an abstraction from reality, but it captures several fundamental
and important aspects of an actual farm system under drought conditions.
100
5
Results and Discussion I: The Base Case Solution
Results and discussion are divided into chapters 5 and 6 for ease of
exposition. Chapter 5 describes and compares the solutions of a binary versus
continuous variables version of the base case model. Continuous variables are
commonly used to approximate a binary variables problem because large
stochastic integer programming models are often difficult to solve. The
implication for the solution of using continuous variables to approximate binary
variables is rarely explored. In this study, both a binary and continuous version of
the base case model can be solved, which presents an opportunity to determine
how closely the solutions resemble one another. Section 5.1 reports and examines
the solution to the binary version of the model. The importance of alternative
drought preparedness and response tools is considered first, followed by a
discussion of the magnitude and variability of drought’s profit impact. Section 5.2
reports the solution to the continuous variables version of the base case model, and
compares it to that of the binary model. Readers are reminded that “optimal”
refers simply, in this dissertation, to activities that are included in the mathematical
programming model’s solution; it does not indicate that the activities are Pareto
optimal or socially efficient.
5.1
The Base Case Solution (Binary Variables Model)
The base case model is representative of recent conditions in the study
area. It is constructed as a six-year stochastic integer programming model, with
two decision stages (fall and spring) in each year. Water supply is known only
probabilistically at the time of fall decisions, and is revealed prior to spring
decisions in each year of the planning horizon. To avoid the influence of a
subjective crop history, the farm’s ten fields (35 acres per field) are assumed to
have no constraining crop history. Sensitivity of the solution to crop history is
reported in section 5.5. Table 5.1 reports the assumed values for several
101
parameters in the base case; tables of other parameter values used in the base case
are reported in Appendix A. Sensitivity of the solution to select parameters is
discussed in chapter 6.
Table 5.1. Parameter values assumed for the base case.
Parameter
Value
Parameter
Water supply
during drought
Value
24 acre-inches
per acre
40 acre-inches
per acre
Interest rate
7%
Discount rate
5%
Full Allotment
Crop History
None
Expected onion
price
$6.00 per cwt
Pr(Drought) in any
given year
40%
Price Stochastic
No
60%
Difference
between the
optimal and
reported solution
< 3.00%
Pr(Full Allotment)
in any given year
The final entry in table 5.1, i.e. “Difference between…,” indicates that the
commercial solution algorithm used to solve the model (CPLEX) generally reports
a solution that only approximates the true optimal solution. This is often the case
for large and complex models, such as the stochastic integer programming model
developed here. The user can define the percentage difference allowed between
the reported solution and the true optimal, including a 0% difference; however,
obtaining a smaller difference often adds hours to the solve time. The approximate
optimal solution, which is within 3% of the true optimal solution for most cases, is
referred to henceforth as the optimal solution, unless otherwise indicated.
The base case solution includes the optimal portfolio of crops, irrigation
technologies, and deficit irrigation levels, as well as discounted profit for 64 water
supply scenarios, and the expected stream of discounted profit. The cropping
activities and profit components of the solution are discussed next.
102
5.1.1 Optimal Cropping Activities
The crop plan that maximizes the expected stream of discounted profit,
given a 40% chance of drought, includes the following activities in the fall of year
1 (figure 5.1): prepare seven fields for onions under drip irrigation, prepare one
field for sugar beets under furrow irrigation, plant one field to winter wheat under
furrow, and plant one field to winter wheat under reuse-furrow. Spring activities
Crops Chosen for Each of 10 Fields
(each field = 35 acres)
Onion D
Wheat F
Wheat F 0.5
Wheat RF
Sg-beet F
Fallow
10
9
8
7
6
5
4
3
2
1
0
Fall
Spr, Full
Spr, Dry
Year 1 Decision Stage, Current Water Supply
Figure 5.1. Optimal fall and spring activities in year 1 of the base case.
Note that spring activities differ for alternative water supply outcomes (full
or dry). Crop Key: F = furrow, RF = reuse furrow, D = drip, 0.5 = 50% of
crop’s irrigation requirement is provided.
depend on whether the water supply outcome (allotment) is full (40 acre-inches
per acre) or dry (24 acre-inches per acre). Activities for a full allotment include
planting and fully irrigating onions and sugar beets in the fields prepared for them,
and fully irrigating wheat. Activities for a dry spring include planting and fully
irrigating onions in the fields prepared for them, fallowing the field prepared for
103
sugar beets, and deficit irrigating the furrow-irrigated wheat (meeting 50% of its
crop water requirement).
Activities prescribed for year 2 are summarized in figure 5.2. The set of
three bars on the left indicates optimal year 2 activities given a full water allotment
in year 1, including individual recommendations for a full versus dry year 2. The
set on the right indicates optimal year 2 activities given a dry year 1, again
including individual recommendations for a full versus dry year 2. Activities
prescribed for year 3 are presented in the same manner in figure 5.3. Graphical
presentation of optimal activities for years 4 through 6 quickly becomes
unmanageable (e.g. year 6 results would require 32 sets of bars), so results for
Sg-beet F
Open
Fallow
Spr, {D} Full
Spr, {D} Dry
Wheat F 0.9
Spr, {F} Dry
Wheat F
Spr, {F} Full
10
9
8
7
6
5
4
3
2
1
0
Fall, {F}
Crops Chosen for Each of 10 Fields
(each field = 35 acres)
Onion D
Fall, {D}
these years are provided in an Excel file located on the attached floppy disk.
Year 2 Decision Stage, {Past} Current Water Supply
Figure 5.2. Optimal fall and spring activities in year 2 of the base case.
Crop Key: F = furrow, D = drip, 0.9 = 90% of crop’s irrigation requirement
is provided.
Spr, {DD} Dry
Spr, {DD} Full
Gcorn F
Fall, {DD}
Fallow
Spr, {DF} Dry
Spr, {DF} Full
Open
Fall, {DF}
Sg-beet F
Spr, {FD} Dry
Spr, {FD} Full
Wheat RF
Fall, {FD}
Spr, {FF} Dry
Spr, {FF} Full
Wheat F
10
9
8
7
6
5
4
3
2
1
0
Fall, {FF}
Crops Chosen for Each of 10 Fields
(each field = 35 acres)
104
Year 3 Decision Stage, {Past} Current Water Supply
Figure 5.3. Optimal fall and spring activities in year 3 of the base case.
Crop Key: F = furrow, RF = reuse furrow.
Note that the activities prescribed for years 2 through 6 are only an
approximation of the optimum. A producer would technically re-solve the sixyear planning problem each year, taking into consideration the outcomes of the
previous years, as well as the effects of the current decision on the next six years.
To mimic this behavior, the model would have to be re-solved for every possible
past water supply outcome. For example, the model would be re-solved twice to
determine optimal year 2 activities for a dry versus full year 1. Re-solving the
model to determine optimal activities for years 3 through 6 is cumbersome because
the number of past water-supply scenarios increases exponentially. The model
would have to be re-solved sixty-four times to generate the conditional set of
optimal activities for year 6. The approximate solution obtained for years 2
through 6 is sufficient for the purposes of this study.
105
The first objective of this research is to determine the role of drought
preparedness versus response in an optimal farm plan. Recall that drought
preparedness techniques are implemented in the fall, before the producer knows
whether the upcoming spring will be full or dry. Drought preparedness reduces
the potential for loss in the event of a drought, and is therefore a form of selfprotection (Ehrlich and Becker 1992). Drought response techniques, in contrast,
are implemented in the spring, after the producer knows that the spring will be dry.
Drought response reduces the magnitude of loss during a drought, and is therefore
a form of self-insurance (Ehrlich and Becker 1992). The following two drought
preparedness techniques appear in the base case solution: using relatively efficient
irrigation technologies on some crops, and leaving some fields open in the fall.
The following two drought response techniques also appear in the base case
solution: fallowing fields, and deficit irrigating crops. These techniques are not
used exclusively in anticipation of, or in response to drought. For example, a
producer who knows that the next six years’ water allotments will be full does
leave some fields open, fallow a field occasionally, and uses relatively efficient
irrigation technologies (figure 5.4). However, these techniques are used more
intensively when the possibility of drought exists (figure 5.4). Each technique is
discussed next.
106
Onion D
Wheat F
Wheat F 0.9
Wheat RF
Sg-beet F
Open
Fallow
Gcorn F
10
8
7
6
5
4
3
2
Fall Yr6, 6F
Spr Yr6, 6F
Fall Yr6, 6D
Spr Yr6, 6D
Fall Yr5, 6F
Spr Yr5, 6F
Fall Yr5, 6D
Spr Yr5, 6D
Fall Yr4, 6F
Spr Yr4, 6F
Fall Yr4, 6D
Spr Yr4, 6D
Fall Yr3, 6F
Spr Yr3, 6F
Fall Yr3, 6D
Spr Yr3, 6D
0
Fall Yr2, 6F
Spr Yr2, 6F
Fall Yr2, 6D
Spr Yr2, 6D
1
Fall Yr1, 6F
Spr Yr1, 6F
Fall Yr1, 6D
Spr Yr1, 6D
Crops Chosen for Each of 10 Fields
(each field = 35 acres)
9
Decision Stage and Year, Certain Water Supply Scenario
Figure 5.4. Optimal fall and spring activities when the producer knows that the next six years’ water allotments will be full
(6F) versus dry (6D). Crop Key: F = furrow, RF = reuse furrow, 0.6 = 60% of crop’s irrigation requirement is provided,
0.5 = 50% provided.
106
107
5.1.1.3 IRRIGATION TECHNOLOGY EFFICIENCY
One form of drought preparedness that appears in the optimal solution is
the use of relatively efficient irrigation technologies. Efficiency refers here to the
proportion of delivered water that is available to the crop for use (i.e. that remains
in the root zone) (table A.4). Irrigation technologies with relatively high
efficiency typically cause less runoff, evaporation, and deep percolation, and
therefore decrease the volume of water that must be delivered to meet crop water
requirements. Irrigation technologies from which the model can choose include
furrow, reuse furrow, wheel line sprinkler, solid set sprinkler, center pivot
sprinkler, and subsurface drip. Readers are likely familiar with most technologies,
with the exception of reuse furrow, which is a modified furrow irrigation system,
in which runoff from the field is collected in a small pond, filtered, and pumped
through pipelines to the top of the field or a nearby field for reuse (Hart et al.
1980). A producer can reduce a crop’s irrigation requirement by using more
efficient irrigation technology. This reduces the probability of a shortage during
dry years, and increases the set of feasible crop combinations during full years.
However, more efficient systems are also more expensive (table A.2). The
expected benefit and cost of investing in more efficient irrigation systems to
increase total crop production or to reduce the likelihood of a water shortage must
therefore be weighed.
The optimal solution suggests growing onions under drip irrigation only.
Drip irrigation is highly efficient (90%), which is beneficial during a water
shortage; however, it also supplies water more uniformly, across time and space, to
the root zone, which increases onion yields. Drip irrigation, therefore, is likely
chosen for reasons other than its water-saving property. That the technology is
also chosen when there is no possibility of drought confirms this (figure 5.4).
Sensitivity analyses are conducted to test whether drip irrigation’s water-saving
property plays any role in its selection, as compared to its yield-enhancing
108
property. Specifically, the yield and cost of onions under furrow and reuse furrow
are set equal to drip irrigation’s, such that the technologies differ only in their
efficiency. The model is re-solved, and drip irrigation is still chosen over furrow
or reuse furrow (this result holds whether drought is certain or uncertain),
suggesting that higher efficiency is valuable.
To rule out multiple optima, the model is further modified by disallowing
drip irrigated onions, and then re-solved. Onions under reuse furrow irrigation are
selected over furrow irrigation, and profit (or expected profit in the case of
uncertainty) is less than when drip irrigation is allowed. This confirms that the
water-saving property of drip irrigation is valuable, not just its yield-enhancing
property. Producers might use drip irrigation on onions largely to increase yield;
however, the technology also enhances profit by reducing onion’s irrigation
requirement, thereby freeing up water for other crops in a full year, and potentially
decreasing the need to deficit irrigate or abandon fields in a dry year. Note,
however, that drip irrigation might also increase the number of fields abandoned or
deficit irrigated in a dry year, particularly if the water savings are sufficient to
support additional fall-prepared or planted crops in a full year (thus causing
additional fields to be attempted), but insufficient in a dry year (thus causing those
additional attempted fields to be abandoned).
It is interesting to note that although producers in the study area grow
onions predominantly under furrow irrigation, drip irrigation is increasingly being
adopted. Additionally, one producer recently updated their drip system such that
the associated pumps, pipes and filters could be moved from year to year. The
yield benefits of drip irrigation were the focus of this producer’s comments,
however, they did indicate that the drought in the early 1990s spurred them to first
consider drip irrigation.
Reuse furrow technology, which is 80% efficient, is also suggested for a
portion of the base case solution’s wheat acreage. Although it is also used when
109
there is no possibility of drought (figure 5.4), the technology is more prevalent
when there is a positive probability of drought (figures 5.1 through 5.3). This
suggests that reuse furrow serves, in part, as a drought preparedness tool. The
ability of reuse technology to reduce water supply risk has been established for
other farm systems (Ziari and McCarl 1995). The above result suggests the same
for an irrigated row crop system. Producers also indicate a recent increase in the
use of reuse furrow systems in the study area. It is unclear, however, to what
degree this is attributable to the recent drought, versus cost-sharing programs
aimed at improving water quality in local waterways.
A numerical example illustrates the water-savings that wheat under reuse
furrow generates, as compared to wheat under furrow irrigation. Wheat under
furrow irrigation requires 40.2 inches of water to meet its seasonal crop water
requirement of 24.1 inches, assuming 50% efficiency and 4 inches of effective
precipitation during the growing season. Wheat under reuse furrow, in contrast,
requires only 25.1 inches to meet its seasonal requirement, because it is 80%
efficient. The use of reuse furrow for an acre of wheat, rather than furrow
irrigation, frees up 15.1 acre-inches for some other use. Alternatively, it decreases
the need to deficit irrigate or abandon wheat during a dry year.
Note that the gains from adopting more efficient technology would be less
if a portion of water “lost” during furrow irrigation actually supported other fields
via subsurface irrigation or return flow (Green and Hamilton 2000). For
simplicity, this is assumed not to occur. It is also assumed that the producer can
replace any field’s irrigation system with another, annually, without incurring a
transaction cost. Specifically, a producer is presumed to be able to sell the old
irrigation system for exactly the balance on the original investment. This
assumption provides more flexibility than is available in reality, and therefore
provides a lower bound on the impacts of drought. Improving the realism of this
feature would require if-then relationships that track and assign costs when a
110
field’s irrigation technology is changed; however, this causes endogeneity in the
existing modeling framework. One alternative for this assumption is to choose an
irrigation system for each field in the first year, and then require this system to
remain in place for the entire planning horizon. This more restrictive assumption
would provide an upper bound on the impacts of drought. It would not, however,
reflect the current use of reuse furrow and drip irrigation in the study area.
5.1.1.4 OPEN FIELDS IN THE FALL
A second form of drought preparedness that appears in the base case
solution is to leave some fields open (i.e. neither prepared nor planted) in the fall
(figures 5.2 and 5.3). Leaving fields open in the fall creates flexibility in the crop
plan; specifically, the producer can accommodate a full or dry spring without
incurring sunk costs, which are generated when a field is fall-prepared or fallplanted (table A.2). Fields left open in the fall can either be allocated to a springprepared and planted crop (e.g. barley, grain corn, or silage corn) if the spring
water allotment is full (figures 5.3 and 5.4), or left fallow if the spring is dry
(figures 5.2 through 5.4). Fields that are prepared or planted in the fall (i.e. not left
open) can also be fallowed if the spring is dry; however, sunk costs generate no
return if the field is fallowed.
The drought literature commonly eludes to the importance of production
flexibility as a drought preparedness strategy (Clawson et al. 1980; Lomas 2000;
Thompson et al. 1996); flexibility is an important concept in the broader
uncertainty literature as well (e.g. Albers 1996). Few studies elaborate, however,
on the specific means by which flexibility can be built into a farm system. A
notable exception is Weisensel et al. (1991), who explicitly examines the value of
flexibility in a dryland wheat-fallow system. The base case solution reveals, for an
irrigated row crop system, that leaving fields open in the fall is an optimal means
to achieve production flexibility. Observations of the study area reveal that
111
producers frequently use this drought preparedness tool in their year-to-year
operations.
5.1.1.5 FALLOWING
Fallowing is the first of two drought response tools considered in this
study. Fallowing can take the following two forms: 1) abandoning a field that was
prepared or planted in the previous fall, or 2) leaving an open field bare.
Fallowing is recommended frequently in the base case solution, particularly when
high-value crops are planned and a water shortage occurs. Fallowing reduces crop
acreage and thus decreases the farm’s total irrigation requirement; it also frees up
water from one field for use in another. Fallowing is an important drought
response tool because the per-acre water allotment during a dry year (24 acreinches) does not meet the per-acre net irrigation requirement of several crops,
including onions (28 acre-inches, assuming 90% efficiency of drip irrigation and
4” of effective precipitation, see table A.8). Model results indicate that producers
should take advantage of this drought response tool; observations from the study
area indicate that producers do. Several survey-based studies of producer
decisions during drought also identify fallowing as a common drought response
tool (Rich 1993; Schuck, Frasier, and Webb 2003; Zilberman et al. 2002).
Figures 5.1 through 5.3 show that, in the event of a drought, open fields are
fallowed first, followed by sugar beet fields, and then wheat fields. Sugar beets
are fallowed before wheat, even though beets are more profitable than wheat. The
choice to abandon one crop rather than another therefore depends on parameters
other than profitability, specifically, available water. The following example
illustrates this point. Cropping activities in either of the “dry year 2” scenarios
(figure 5.2) leave 137 acre-inches of excess water. Suppose that after a drought is
revealed in year 2, a producer decides to modify the optimal solution. Instead of
fallowing sugar beets and keeping wheat, they decide to keep one field of sugar
beets under furrow irrigation, and fallow one field of wheat under furrow
112
irrigation. The quantity of water saved by fallowing the wheat field satisfies only
70% of the beet’s water requirement, and deficit irrigated sugar beets are less
profitable than fully irrigated wheat. Suppose the producer instead keeps one field
of sugar beets under reuse furrow. Sugar beets under reuse furrow are also less
profitable than wheat under furrow or reuse furrow. The rational choice for the
producer, given a water shortage of the magnitude modeled here, is to fallow sugar
beets, rather than wheat.
Note that fallowing during a full year is limited in the model to one field or
less. This constraint forces crops that generate small or negative profit directly,
but that contribute to more profitable crops though soil-quality enhancement and
nutrient management, into the plan. Production functions that quantify these and
other crop inter-dependencies are not readily available, nor are data of sufficient
detail or duration available to estimate functions for the study area. In addition,
agronomic constraints included in this model do not capture completely the true
production function. Constraints, such as limited fallowing during a full year, are
therefore used in lieu of the true production function. These constraints primarily
prevent unrealistic solutions, such as growing only potatoes and onions, and
fallowing the land for years until it is again eligible for these crops.
5.1.1.6 DEFICIT IRRIGATION
Deficit irrigation is the deliberate and systematic under-irrigation of crops
(English and Raja 1996). The goal of deficit irrigation is to expose plants to
controlled levels of water stress in order to conserve water without causing
significant yield reductions (Kirda 2002). Alternatively, deficit irrigation can be
used to equilibrate the marginal value of water across crops. That is, profit might
be increased by deficit irrigating a crop with low marginal value of water (even if
it reduces yield significantly) to provide water to a crop with high marginal value
of water (English and Nakamura 1989; Kirda 2002). Two types of deficit
irrigation can be practiced: season-long deficit irrigation, in which the crop is
113
deficit irrigated by the same proportion throughout the entire growing season, and
strategic deficit irrigation, in which the crop is deficit irrigated only during its most
drought-tolerant growth stages (Kirda, Kanber, and Tulucu 1999).
Strategic deficit irrigation generally results in smaller yield reductions than
season-long deficit irrigation (Bazza 1999; Hargreaves and Samani 1984);
therefore, producers are likely to use the strategic rather than season-long
approach. However, a linear yield response function for strategic deficit irrigation
is not readily available; in contrast, a function for season-long deficit irrigation
exists and is well-established in the literature (Doorenbos and Kassam 1979).
Season-long deficit irrigation is therefore used in this dissertation, and table A.6
reports the effect of alternative deficit irrigation levels on per-acre yield and profit.
Deficit irrigation may be used more frequently, in practice, than the optimal
solution indicates, if producers use strategic deficit irrigation rather than seasonlong deficit irrigation. Note, however, that Bazza (1999) indicates that a producer
who is uncertain of the most critical growth stage in which to irrigate might do
better to practice season-long deficit irrigation to achieve water conservation goals.
Deficit irrigation, the second drought response tool considered in this
study, appears in the base case solution. Wheat, in particular, is the primary crop
that is deficit irrigated. This is attributable to wheat’s relative drought tolerance
(its yield response coefficient (table A.5) is not greater than 1), its low cost of
production (table A.2), and its relatively low market value (table A.1). Wheat, in
summary, is one of the few crops that remain profitable under season-long deficit
irrigation (table A6). Additionally, wheat has a relatively low total irrigation
requirement (table A.3), which enables the producer to more frequently support
deficit irrigated wheat during years in which the quantity of excess water is small,
as compared to more water-demanding crops.
The above result is consistent with existing studies that show winter wheat
to be well-suited for, and profitable under deficit irrigation in many cases (Bazza
114
1999; English 1990a; Musick and Dusek 1980). More generally, it supports the
findings of Bernardo et al. (1987) and English (1990b), who report, respectively,
that producers in Washington’s Columbia River Basin (an area of similar climate
and farming systems) should and do practice deficit irrigation during a water
shortage. Producers in the study area also indicate that they deficit-irrigate less
sensitive and less valuable crops to ensure water for more sensitive and more
valuable crops, or to increase the proportion of fields planted. It is not known,
however, how common the use of deficit irrigation is among producers in the
study area. Zilberman et al. (2002) and Schuck et al. (2003) report, respectively,
that only a small portion of California and Colorado producers used deficit
irrigation during past droughts.
The literature also indicates minimal yield loss under deficit irrigation for
sugar beet (Kirda, Kanber, and Tulucu 1999); however, sugar beet is rarely deficit
irrigated in the base case solution. This corresponds with Bazza’s finding (1999)
that maximum profit from sugar beet is obtained when the crop’s water
requirement is fully met. Nonetheless, under the model’s assumed parameters,
sugar beet is profitable if 80% or more of their irrigation requirement is met. It is
not clear then why sugar beet is not deficit irrigated in the optimal solution. One
possible explanation is that sugar beet requires a relatively large quantity of water,
particularly in comparison to wheat. Sugar beet might therefore have to be
severely deficit irrigated to generate sufficient water savings during a drought.
Severe deficit levels are associated, in the model, with unprofitable sugar beet
yields. Another possible explanation is that results from the literature are largely
based on strategic deficit irrigation, whereas the model’s results are based on
season-long deficits. The season-long representation used in the model likely
overestimates yield losses associated with strategic deficit irrigation (Fujun et al.
1999), in which case deficit irrigation will be recommended less under season-long
deficit irrigation than it would be under strategic deficit irrigation. Finally, one
115
benefit of deficit irrigation that is highlighted throughout the literature is the ability
to expand crop acreage using the conserved water (English 1990a). The producer
in the model is unable to expand farm acreage beyond the 350 owned acres; the
benefit of deficit irrigation to which the literature refers might therefore be larger
than that captured in the model.
To summarize the above results regarding optimal cropping activities, the
base case solution indicates a role for both drought preparedness tools (leaving
fields open and using relatively efficient irrigation technologies) and drought
response tools (fallowing and deficit irrigation). Observations of producers in the
study area validate this result; many of the preparedness and response activities are
currently used. Specifically, producers commonly leave fields open to provide
production flexibility, deficit irrigate less sensitive crops, and fallow fields in
response to drought. More efficient irrigation technologies are also becoming
more common in the study area, in particular, reuse furrow and drip irrigation.
The preparedness and response tools discussed above have been analyzed in
previous studies. However, few studies have considered them simultaneously.
The above results indicate that all of the tools are needed to optimally prepare for
and respond to drought. The results also illustrate the degree to which a riskneutral producer in an irrigated row crop system should use each tool.
5.1.2 Profit Outcomes
Profit outcomes associated with the base case solution are presented in
table 5.2, figure 5.5, and table A.7. Recall from chapter 4 that “profit” refers, in
this study, to returns to land and management. Table 5.2 and figure 5.5 describe
the distribution of the stream of discounted profit for the six-year planning
horizon. These will be compared in subsequent sections with profit outcomes
from alternative versions of the model. The profit impact of drought, particularly
as it varies with the duration of drought, is the focus of this section. The profit
impact of drought is defined as the difference in discounted profit between a
116
scenario that includes drought and a scenario that does not (e.g. [Dry Full Full Full
Full Full] versus [Full Full Full Full Full Full] for a single-year drought of interest,
or [Dry Dry Full Full Full Full] versus [Full Full Full Full Full Full] for a multiyear drought of interest).
Table 5.2. Summary statistics of the base case solution’s profit outcome.
Statistic
Value ($)
Expected Stream of Discounted Profit
531,853
Standard Deviation of Expected Stream
36,076
Maximum Discounted Profit
590,100
Discounted Profit of Scenario
582,703
[Full Full Full Full Full Full]*
Minimum Discounted Profit
408,273
*
An explanation for the discrepancy between maximum discounted profit
and discounted profit for this scenario is provided at the end of section
5.2.1.
Proportion of water supply scenarios
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
380
400
420
440
460
480
500
520
540
560
580
600
620
Discounted Profit (thousand $)
Figure 5.5. Cumulative distribution function of the stream of discounted
profit for the base case solution. E.g. approximately 70% of water supply
scenarios (n=64) generate less than $540 thousand in discounted profit.
117
The primary objective of this section is to determine whether the impacts
of drought can be generalized in a useful manner for policymakers. For example,
it is not clear whether a three-year drought can be presumed to generate a larger
profit loss than a two-year drought. In reality, the characteristics of individual
droughts are sufficiently unique to make comparison difficult. The model, in
contrast, assumes a single level of severity for every drought (24 acre-inches per
acre, rather than 40), such that the affect of duration on profit loss can be analyzed.
Figure 5.6 reports the average loss of discounted profit for droughts of various
duration (from one year to six years), as well as the minimum and maximum profit
loss for each duration category. Average profit loss is reported because a drought
of particular duration (e.g. a single-year drought) can occur at alternative points in
the six-year planning horizon (e.g. [Dry Full Full Full Full Full] versus [Full Full
Full Full Full Dry]). The profit impact of drought varies depending on the point in
the crop plan at which it occurs (Appendix H), because crop plans vary across
years. Drought that occurs during a year in which many fields are left open in the
fall, for example, has less impact than one that occurs during a year in which all
fields are fall-prepared.
Three features of figure 5.6 indicate that few generalizations can be made
about the profit impact of drought by duration alone. First, average profit loss
increases as the number of droughts during the planning horizon increases.
However, the magnitude of profit loss increases at a decreasing rate. Two years of
drought cause 183% larger losses than one year of drought. Three years of
drought cause 68% larger losses than two years of drought; four years of drought
cause 48% larger losses than three years; five years cause 38% larger losses than
four years, and six years cause 33% larger losses than five years. The marginal
impact of each additional drought is smaller. This result stems from the fact that
the impacts of drought are larger for some scenarios than others. As the number of
years of drought experienced increases, the probability increases that the worst-
118
impacted scenarios have already been experienced. Hence, the marginal impact of
Change in total discounted profit ($1000)
an additional year of drought is likely to be small on average.
20
10
0
-10
-20
-30
-40
-50
-60
-70
-80
-90
-100
-110
-120
-130
-140
-150
-160
-170
-180
-190
1
2
3
4
5
6
Years of Drought
Figure 5.6. Average change in total discounted profit (black dashes) by
years of drought experienced (as compared to 6 years of full water supply).
Gray brackets indicate the maximum and minimum impact of drought.
E.g. Three years of drought can cause a minimum loss of $29,000 in
discounted profit, or a maximum loss of $96,000, depending on the years
in which the droughts occur. Discounted profit loss, on average, is $64,000
when droughts occur in three years out of six.
Second, profit loss within each drought category varies widely, with the
standard deviation of loss generally increasing as the number of droughts increases
(table 5.3). An example illustrates the importance of this feature. The largest
profit loss attributable to three years of drought is $96,000, which occurs in the
scenario [Full Dry Dry Full Dry Full]. The smallest profit loss attributable to three
years of drought is $29,000, which occurs in the scenario [Full Dry Full Full Dry
119
Dry]. A drought in year 3, rather than year 6, more than triples profit loss. The
policy implication of this result is that the number of droughts alone cannot
precisely predict the magnitude of profit losses. In fact, as the number of droughts
increases, profit loss becomes more variable and harder to generalize. The timing
of drought events within the planning horizon (i.e. within the crop plan) is a
crucial determinant of profit losses. Unfortunately, crop plans are likely to vary
widely across producers; even producers who implement the same six-year crop
plan could be in different years of the plan when a drought occurs. Profit losses
attributable to a particular drought event are therefore likely to vary widely across
producers.
Table 5.3. Characteristics of change in total discounted profit ($) as the
years of drought experienced (during the 6-year planning horizon)
increases.
Years of
Drought
1
2
3
4
5
6
Avg. Change Max. Change
in Disc. π*
in Disc. π
-13,488
-28,106
-38,216
-68,491
-64,198
-95,840
-95,109
-120,484
-131,145
-145,230
-174,430
-174,430
Min. Change
in Disc. π
7,397
-6,499
-28,543
-57,427
-113,777
-174,430
Std. Dev.
of Change
13,502
16,600
18,484
19,246
12,082
0
*
Change in discounted profit is calculated as discounted profit of the
drought scenario less the discounted profit of scenario [Full Full Full Full
Full Full]. Negative numbers indicate profit loss.
The discount rate, interest rate, and differences in the crop plans of years 3
and 6 explain how the timing of drought influences the resulting profit loss. The
discount rate implies that losses in year 3 are given more weight than losses in year
6. The interest rate implies that losses in year 3 will reduce earned interest (on
savings) for more years than losses in year 6. Finally, fall crop plans in place prior
to the year 3 drought versus the year 6 drought lead to larger losses in the year 3
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drought. Specifically, one field is left open in the fall of year 3 (scenario [Full
Dry]), while four fields are left open in the fall of year 6 (scenario [Full Dry Full
Full Dry]). Drought in year 3 forces the producer to fallow four fields, including
three fields prepared for sugar beets. Drought in year 6 also forces the producer to
fallow four fields, but all were open, so no fall investments were lost.
A third feature of figure 5.6 that has important implications for
generalizations about drought is that the maximum loss attributable to two years of
drought can exceed the average loss attributable to three years of drought. The
same holds for three years of drought, whose maximum loss can exceed the
average loss of four years of drought. This feature highlights the difficulty of
generalizing profit loss attributable to drought. Profit loss grows larger, on
average, as the number of droughts increases, but in some cases, fewer years of illtimed drought can cause greater losses than more years of favorably-timed
drought.
In conclusion, few generalizations can be made about the profit impact of
drought based on its duration alone. Specifically, policymakers should not view
droughts within the same duration category equally, or necessarily base their
expectations of loss for a drought on the impacts of similar past droughts. A
producer could experience multiple identical droughts throughout their life, and be
affected differently by each of them, depending on the crops in place when the
droughts occur. Similarly, two producers that follow the same crop plan, but are in
different stages of that plan, could experience the same drought and yet incur
considerably different profit losses. This result highlights the difficulty that
policymakers face when determining the need for, and appropriate extent of,
assistance during or after a drought. The magnitude of impact might be highly
variable across producers, even in a homogeneous agricultural area. Policymakers
would also be ill-advised to assume that the profit impact of a relatively short-lived
drought is less than that of a more prolonged drought. While this is true on
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average, there is considerable overlap of the ranges of profit loss for droughts of
various durations.
The base case solution raises several questions that parametric or structural
variations of the base case model can be used to address. These questions are
raised and addressed in the remaining sections of chapter 5 and in chapter 6.
Before proceeding, however, readers may have noted from tables 5.2 and A.7 that
the scenario “six full years” does not generate the maximum discounted profit.
This implies that a scenario containing drought generates more profit than the
scenario with no drought. The reason for this initially counterintuitive result is
explained here. Fallowing is limited in the model to one field or less during a full
year; in contrast, fallowing is unlimited in a dry year. Several crop-irrigation
combinations in the model generate negative returns to management and land
under the assumed prices, yields, and costs (including a 7% opportunity cost of
money). Grain corn under reuse furrow, for example, generates -$38.55 per acre.
However, from a whole farm budget perspective, they contribute to the
sustainability of long-term production of profitable crops, such as onions, by
utilizing excess nutrients, reducing pests and disease, and maintaining soil quality.
The model does not capture these benefits, however, and therefore underestimates
the economically profitable level of such crops. To counter-balance the systematic
under-estimation of these crops’ economic benefits, the model is required to plant
most acres during a full year, regardless of whether the crops brought into solution
are economically profitable on that field in that year.
Fallowing is not limited during dry years, however, because fallowing is
needed to ensure water for high-value crops. Crops that generate small or negative
profit are fallowed first. Fallowing crops that generate negative profit (e.g. grain
corn under reuse furrow) causes profit in the drought scenario to be higher than
that in the full scenario. The impact on most scenarios’ profits is relatively small,
however, typically around $1000. A reader might infer that fields are fallowed
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during dry years for profit-purposes only, rather than water-management purposes.
This does not appear to be the case. Excess water (i.e. the volume of allocated
water less the volume required for a specific crop plan) is insufficient in all 63
drought scenarios to support an additional field of any crop. That is, it is almost
always a physical necessity, not just profitable, to fallow rather than plant an
unprofitable crop during a drought.
5.2
The Base Case Solution (Continuous Variables Model)
Many farm-level linear programming models define decision variables as
continuous, rather than integer, to enhance problem solvability. The results
reported in this dissertation are primarily from an integer (binary) model, which
enables crop history for each field to be tracked through time, and hence
agronomic constraints to be enforced at the field-level, rather than at the farmlevel. The binary variables model is a more accurate representation of production
in the study area; however, it requires a more powerful solution algorithm and
more time to solve than a continuous variables model. The solution to a
continuous variables version of the base case model is presented in this section,
and the similarities and differences between the continuous and binary models’
solutions are discussed.
Figures 5.7 through 5.9 report the optimal crop plan for years 1 through 3
of the continuous variables model. Similarities to the optimal crop plan for the
binary variables model include the following: 1) most of the acres are planted to
onions under drip irrigation in the first year, with the remaining eligible acres
planted in year two; 2) reuse furrow irrigation is used on a portion of the wheat
acreage, 3) some acres are left open in the fall, and 4) aside from open fields, sugar
beets are the first crop fallowed during drought. The continuous model’s optimal
solution differs, however, in the following ways: 1) deficit irrigation is not
included, 2) more onions are planted in the first year, and 3) more wheat is planted
under reuse furrow and less under furrow irrigation. The profit outcomes for the
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continuous variables model (table 5.4 and figure 5.10) are similar, however, to the
binary variables model. Differences in the optimal solutions do not lead to
significantly different profit outcomes, in this case.
Crops Chosen for Each of 350 Acres
Onion D
Sg-beet F
Fallow
350
300
250
200
150
100
50
0
Fall
Spr, Full
Spr, Dry
Year 1 Decision Stage, Current Water Supply
Figure 5.7. Optimal fall and spring activities in year 1 of the continuous
variables model’s base case solution. Note that spring activities differ for a
full versus dry spring. Crop Key: F = furrow, D =drip.
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Crops Chosen for Each of 350 Acres
Onion D
Wheat F
Wheat RF
Open
Fallow
Sg-beet F
350
300
250
200
150
100
50
0
Fall, {F}
Spr, {F}
Full
Spr, {F}
Dry
Fall, {D} Spr, {D} Spr, {D}
Full
Dry
Year 2 Decision Stage, {Past} Current Water Supply
Figure 5.8. Optimal fall and spring activities in year 2 of the continuous
variables model’s base case solution. Crop Key: F = furrow, RF = reuse
furrow, D =drip.
Sg-beet F
Open
Fallow
Gcorn F
300
250
200
150
100
Spr, {DD} Dry
Spr, {DD} Full
Spr, {DF} Full
Fall, {DF}
Spr, {FD} Dry
Spr, {FD} Full
Fall, {FD}
Spr, {FF} Dry
0
Spr, {FF} Full
50
Fall, {FF}
Crops Chosen for Each of 350 Acres
Wheat RF
Fall, {DD}
Wheat F
350
Spr, {DF} Dry
125
Year 3 Decision Stage, {Past} Current Water Supply
Figure 5.9. Optimal fall and spring activities in year 3 of the continuous
variables model’s base case solution. Crop Key: F = furrow, RF = reuse
furrow.
Table 5.4. Summary statistics of the base case’s profit outcome for the
continuous variables model.
Statistic
Value ($)
Expected Stream of Discounted Profit
547,049
Standard Deviation of Expected Stream
36,436
Maximum Discounted Profit
597,958
Minimum Discounted Profit
416,793
Change in total discounted profit ($1000)
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0
-10
-20
-30
-40
-50
-60
-70
-80
-90
-100
-110
-120
-130
-140
-150
-160
-170
-180
-190
-200
1
2
3
4
5
6
Years of Drought
Figure 5.10. Average change in the continuous variables model’s total
discounted profit (black dashes) by years of drought experienced (as
compared to 6 years of full water supply). Gray brackets indicate the
maximum and minimum impact of drought.
The most noteworthy difference in the solutions is that deficit irrigation
appears in the binary model’s optimal solution, but not in the continuous model’s.
This is because the binary model constrains producers to make discrete decisions,
specifically 35 acres at a time. The continuous model, in contrast, allows the
producer to make continuous acreage decisions, which allows more flexible
allocation of resources. A producer could choose in the continuous model, for
example, to furrow irrigate 18.6 of their 350 acres. The equivalent choice in the
binary model, in contrast, is to furrow irrigate 35 acres (1 field), or no acres.
Suppose the producer chooses, in the binary model, to furrow irrigate 1 field.
They might have to use deficit irrigation, in the event of drought, to compensate
for the “extra” 16.4 acres of relatively inefficient furrow irrigation. The producer,
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in the continuous model, can apply a more efficient irrigation technology to the
16.4 acres, and possibly eliminate the need for deficit irrigation. Therefore, a
modeler who uses a continuous variables model to represent a binary variables
farm system might conclude erroneously that deficit irrigation is unnecessary.
The farm system being modeled should influence whether a binary or
continuous variables model is used. Many farm landscapes are divided into
discrete fields by windbreaks, streams, and roads. It is impractical, in some cases,
to apply multiple crops or irrigation technologies to an individual field, or to
change the size of a field, at least in the short-run. These producers are likely to
choose one crop and irrigation technology per field. A binary variables model
would best represent this decision environment. How closely a continuous
variables model can approximate this is likely to decrease as field size increases
(again, assuming the producer can choose only one crop and technology per field,
regardless of field size).
For example, suppose a farmer has two 100-acre fields to which they can
apply only one crop each. A binary variables model might suggest allocating one
field to wheat, and the other to sugar beets. A continuous variables model might
suggest allocating one-fifth of the acreage (0.4 fields) to corn, three-fifths (1.2
fields) to sugar beets, and one-fifth to wheat. While it would be clear from the
continuous model’s solution that the (more accurate) binary model would allocate
at least one field to sugar beets, it would be less clear which crop it would allocate
to the second field. The continuous model might also identify more than two
irrigation technologies, making it unclear which technology a binary model would
assign to each field. As a result of these tendencies, the continuous model might
also misestimate profit outcomes. Smaller field sizes would likely reduce the
difficulty of translating a continuous model’s solution into a binary (field-by-field)
solution.
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In summary, it is unlikely that a continuous variables model will accurately
represent the farm system being studied when taking scale characteristics such as
field size into consideration. However, the availability of a solution algorithm that
is capable of solving more complex binary models, as well as time limits, should
also influence the choice of a continuous versus binary model.
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6
Results and Discussion II: Applications of the Binary Model
Chapter 6 is the second of two chapters devoted to results and discussion.
The binary variables version of the base case model is used in this chapter to
explore several aspects of optimal drought preparedness and response given water
supply uncertainty and inter-year dynamics. Readers are again reminded that
“optimal” refers simply, in this dissertation, to activities that are included in the
mathematical programming model’s solution; it does not indicate that the activities
are Pareto optimal or socially efficient. Section 6.1 explores the role of the interest
and discount rates in the base case solution. Section 6.2 presents solutions to a
certainty version of the model, which differentiates the impact of anticipated
versus unanticipated drought. Section 6.3 discusses the importance of recognizing
inter-year dynamics when estimating the impacts of drought on cropping activities
and profit. Section 6.4 investigates the impact of previous crop history on the base
case solution. Section 6.5 introduces price uncertainty into the model and explores
its effects on optimal drought preparedness and response. Section 6.6 explores the
usefulness of the multi-peril crop insurance program’s prevented planting
provision as a drought preparedness tool. Section 6.7 considers the impact of
climate change on optimal drought preparedness and response, and the resulting
profit impact. Specifically, the impacts of increases in the frequency of drought,
severity of drought, and both frequency and severity of drought are considered.
6.1
The Role of Interest and Discount Rates
Agronomic constraints dictate that each field can be planted to onions only
once in a six-year period to avoid disease; hence, a total of 10 fields can be planted
to onions over the six-year planning horizon. The binary variables version of the
base case solution prescribes planting seven of those ten fields to onions in the first
year. This seems counter to the notion of diversification as a drought preparedness
tool (Lomas 2000; Vlachos and James 1983; Yevjevich and Vlachos 1983) and to
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the approach taken by most onion producers in the study area. Onions are the
primary source of profit in the study area; they require a $600 fall investment, and
are among the most water-sensitive crops grown. A more intuitively appealing
plan, from a drought preparedness perspective, would spread the ten fields of
onions out over several years, and include drought-tolerant crops, such as wheat, in
the suite of crops planted each year. This would reduce the likelihood of
abandonment or deficit irrigation of onions during a water shortage. The objective
of this section is to determine what drives the model’s initially counterintuitive
solution, or alternatively to determine what motivates producers in the study area
to spread high-value crops throughout the planning horizon.
Positive interest and discount rates provide incentive to plant valuable
crops first, primarily to generate as much profit as soon as possible. Interest on
profit can thus be earned for a longer period, and the discounting of profit can be
minimized. These incentives could be driving the specialization (as opposed to
diversification) in year 1 of the base case solution. However, water supply
uncertainty, and agronomic constraints that generate inter-year dynamics, also
influence the solution. It is therefore not clear, based on intuition alone, whether
specialization in year 1 of the base case solution is attributable to the discount and
interest rates alone, or also to water supply uncertainty and inter-year dynamics.
Sensitivity of the solution to alternative interest and discount rates is therefore
tested.
The interest and discount rates are reduced from the base case values of 7
and 5%, respectively, to the extreme values of 0 and 0% to test their role in the
base case solution. No onions are planted in the year 1 when the interest and
discount rates are set equal to zero (figure 6.1). A few fields are planted to onions
in years 2 and 3 (figures 6.2 and 6.3). Onions are planted, in fact, throughout the
planning horizon in the absence of an interest or discount rate, in sharp contrast to
the base case. Drought management does not appear to be the motivation for
131
diversification in the absence of discount and interest rates, however. The
motivation is the opportunity to shift crops through time to make use of a larger
proportion of the total water allotment expected over six years, thereby increasing
crop acreage and profit. Details of this opportunity are described next.
Crops Chosen for Each of 10 Fields
(each field = 35 acres)
Wheat F
Wheat F 0.6
Wheat F 0.5
Wheat RF
Sg-beet F
Open
Fallow
10
9
8
7
6
5
4
3
2
1
0
Fall
Spr, Full
Spr, Dry
Year 1 Decision Stage, Current Water Supply
Figure 6.1. Optimal fall and spring activities in year 1 of the “zero interest
and discount rates” case. Crop Key: F = furrow, RF = reuse furrow, 0.6 =
60% of crop’s irrigation requirement is provided, 0.5 = 50% provided.
Onions under drip irrigation require less water than many other crops. The
base case crop plan, which allocates most fields to onions in year 1, leaves 2,800
acre-inches of the year 1 water allotment (in a full year) unused. This quantity of
excess water is sufficient to supply 40 acre-inches to two additional fields. The
producer in this model does not have access to additional fields, however, nor can
they sell or store the excess water. The only means of using this water is to plant
more water-intensive crops in year 1, and shift onion production into future years.
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The benefit of increasing the proportion of the water allotment used is that more
fields can be planted over the six-year planning horizon. The opportunity cost is
the delay of onion profit. Cost exceeds benefit when discount and interest rates are
positive, but not when discount and interest rates are zero. In the latter case, onion
production is therefore shifted from year 1 to years 2 through 6, more waterintensive crops are shifted into year 1, and additional fields can be planted, in most
scenarios, over the planning horizon.
Onion D
Wheat F
Wheat RF
Wheat F 0.8
Sg-beet F
Open
Fallow
Gcorn F
Crops Chosen for Each of 10 Fields
(each field = 35 acres)
10
9
8
7
6
5
4
3
2
1
0
Fall, {F} Spr, {F} Spr, {F}
Full
Dry
Fall, {D} Spr, {D} Spr, {D}
Full
Dry
Year 2 Decision Stage, {Past} Current Water Supply
Figure 6.2. Optimal fall and spring activities in year 2 of the “zero interest
and discount rates” case. Crop Key: F = furrow, RF = reuse furrow, D =
drip, 0.8 = 80% of crop’s irrigation requirement is provided.
133
Spr, {DD} Dry
Fallow
Spr, {DD} Full
Gcorn F
Fall, {DD}
Spr, {DF} Dry
Open
Spr, {DF} Full
Sg-beet F
Fall, {DF}
Spr, {FD} Dry
Spr, {FD} Full
Wheat RF
Fall, {FD}
Wheat F
Spr, {FF} Dry
Fall, {FF}
10
9
8
7
6
5
4
3
2
1
0
Spr, {FF} Full
Crops Chosen for Each of 10 Fields
(each field = 35 acres)
Onion D
Year 3 Decision Stage, {Past} Current Water Supply
Figure 6.3. Optimal fall and spring activities in year 3 of the “zero interest
and discount rates” case. Crop Key: F = furrow, RF = reuse furrow, D =
drip.
A comparison of overall crop activities for the “zero rates” case versus the
base case confirms that, in the absence of interest and discount rates, fall attempts
of wheat and sugar beets increase. Additionally, for many water scenarios, more
wheat and sugar beet fields are successful, i.e. fewer fall-planted fields are
fallowed. The benefit of spreading onions over the planning horizon, in summary,
is that more fields are planted, and more crops are successful. It appears, however,
that the benefits of this strategy accrue mostly during full years. Shifting crops
between years frees up additional water in all years. The model takes advantage
by attempting more fall-prepared or planted crops. The additional water is
sufficient in full years to allow for additional fields to be planted, but is
insufficient in dry years. This leads to higher profits in full years, and higher
expected net crop revenue overall, without an increase in profit variability.
134
However, it also leads to reduced profits in some dry years (relative to the base
case) because the additional prepared or planted fields have to be deficit irrigated
or abandoned. Average profit losses associated with drought for the “zero rates”
crop plan (figure 6.4a) are, therefore, slightly higher for some droughts than those
for the base case crop plan (figure 6.4b).
The absence of interest and discount rates, in summary, decreases the
benefit of specialization in the first year of the crop plan, or equivalently,
decreases the cost of diversification (delayed profit). The benefit of diversification
in year 1 is that more fields can be planted and grown successfully, particularly in
full years, over a six-year period. Expected net crop revenue in the zero rates case
is therefore higher (without an accompanying increase in profit variability) than
the base case. Note, however, that diversification in the presence of positive
discount and interest rates will not increase expected net crop revenue (otherwise
the base case solution would recommend diversification). Surprisingly,
diversification in year 1 does not lessen the average profit impact of drought, and
actually increases it slightly for some droughts. Diversification is therefore not a
form of drought preparedness for the conditions assumed in this model. The
policy implication of this result is that extremely low interest and discount rates
could induce producers to adopt crop plans that result in larger loss during some
droughts.
These conclusions provide insight about some of the tradeoffs that
producers under water supply uncertainty face when deciding whether to
specialize or diversify within a production year. The tendency of producers in the
study area to diversify, counter to the results found here, suggests the presence of
equipment or labor constraints not captured in the model, or other sources of
uncertainty (e.g. price) that make specialization too risky (the topic of section 6.5),
other means to use excess water in year 1, more severe drought than what is
defined here (the topic of section 6.7), or an erroneous compulsion to make full
Change in total discounted profit ($1000)
135
0
-10
-20
-30
-40
-50
-60
-70
-80
-90
-100
-110
-120
-130
-140
-150
-160
-170
-180
-190
1
2
4
3
5
6
5
6
Years of Drought
(a)
Change in total discounted profit ($1,000)
Zero Rates Case
Base Case
20
10
0
-10
-20
-30
-40
-50
-60
-70
-80
-90
-100
-110
-120
-130
-140
-150
-160
-170
-180
-190
1
2
3
4
Years of Drought
(b)
Figure 6.4. (a) Average change in the “zero interest and discount rates”
model’s total discounted profit (gray circles) by years of drought
experienced (as compared to 6 years of full water supply). Gray brackets
indicate the maximum and minimum impact of drought. (b) A comparison
of average change in total discounted profit for the “base case” (black
triangles and brackets) versus “zero rates case” (gray circles and brackets).
136
use of their water allotment despite positive discount and interest rates. Risk
aversion related to water supply uncertainty is not strongly implicated as the cause
of producer’s diversification behavior, since the standard deviation of discounted
profit is nearly the same for the base case versus zero rates case solution ($36,100
versus $33,600).
As a final note, the above results highlight an interesting hypothetical link
between instream flow issues and the effects of interest and discount rates on the
crop plan. There is more excess-water during dry years than during full in the
“zero rates” case, while this is rarely true in the base case. Again, this is
attributable to the fact that the “zero rates” crop plan frees up additional water,
which is then used in full years, but unused in dry years. Instream flows are often
of critical concern during dry years. Extremely low or subsidized interest and
discount rates could ease instream flow issues during dry years and increase them
during full years. This result is highly sensitive, however, to increases in the
interest rate. A 1% interest rate is sufficient to restore much of the base case
solution, such that there is more excess water in full years than dry. Zero interest
and discount rates are highly contrived, so this link likely has limited implications
in practice. The notion that macroeconomic parameters can impact water use via
crop rotation patterns is interesting, however, and perhaps worthy of additional
research.
6.2
The Role of Uncertainty
Uncertainty about the timing and duration of drought exacerbates profit
loss attributable to drought. A producer, who knows in advance when a drought
will occur and how long it will persist, is able to prepare perfectly for the event.
This does not imply that the drought will generate no loss. A water shortage, even
when anticipated, is likely to reduce profit, because fewer crops can be grown.
The cost of uncertainty for a particular scenario can be estimated by comparing
profit in the stochastic model to that in the deterministic model (section 4.5.1).
137
The deterministic model assumes that the water supplies for all six years of the
planning horizon are known in the fall of year 1. This is an extreme assumption,
but provides insights about the impacts of uncertainty. A few scenarios are
presented next to illustrate how optimal drought preparedness and response under
certainty differs from that under uncertainty.
The first scenario analyzed is [Full Full Full Full Full Full]. The objective
is to determine how certainty changes the crop plan and profit outcomes
(compared to the base case crop plan under uncertainty). A producer who knows
in the fall of year 1 that water allotments in the next six years will be full is able to
change the crop plan, such that additional fields of more valuable fall-prepared or
planted crops can be grown, rather than less-valuable spring-prepared and planted
crops or fallowing. Specifically, two additional fields each of sugar beets and
wheat are substituted for three fields of grain corn and one field of fallow (table
6.1). Discounted profit under certainty therefore exceeds that under uncertainty by
$24,568 (a 4% increase).
The producer is able to increase the number of successful fields not only
because a full water supply is guaranteed, but also because they shift the timing of
crops across the planning horizon (figure 6.5) to more fully utilize the certain
water supply each year. Table 6.2 shows that excess water is both smaller and less
variable under certainty than under uncertainty. The primary shift in the timing of
crops is that onions are spread over three years, rather than two, and additional
sugar beets, which are more water-intensive, are planted in year 1. A tradeoff
exists, however, between the benefit and cost of retiming crops. The benefit is an
extra field of sugar beets or wheat, but the cost is the delay of profit from onions.
The benefit outweighs the cost for this particular water supply scenario under
certainty, but it may not for all scenarios, and it does not under uncertainty.
138
Table 6.1. Number of fields successfully planted to various crops in
scenario [Full Full Full Full Full Full] under certainty and uncertainty.
Crop
Onion
Sugbeet
Wheat
Wheat
Gcorn
Gcorn
Fallow
Irrig Deficit
Drip
D1
Furrow
D1
Furrow
D1
Reusefrw
D1
Furrow
D1
Reusefrw
D1
Furrow
D7
# Successful
Fields
Cert Uncert
10
10
13
11
22
19
5
6
6
8
0
1
4
5
60
60
Table 6.2. Excess water (i.e. the quantity of water remaining after crop
water requirements are met) by year for scenario [Full Full Full Full Full
Full] under uncertainty and certainty.
Uncertainty
Certainty
Excess Water (acre-inches)
Yr1 Yr2 Yr3 Yr4 Yr5 Yr6 Total
2,801 1,241
54
54 302 451 4,903
4 969
27 161 213 292 1,666
139
Onion D
Wheat F
Wheat RF
Sg-beet F
Open
Gcorn F
Gcorn RF
Fallow
10
Crops Chosen for Each of 10 Fields
(each field = 35 acres)
9
8
7
6
5
4
3
2
Spr Yr6 C
Spr Yr6 U
Fall Yr6 U
Spr Yr5 C
Spr Yr5 U
Fall Yr5 U
Spr Yr4 C
Spr Yr4 U
Fall Yr4 U
Spr Yr3 C
Spr Yr3 U
Fall Yr3 U
Spr Yr2 C
Spr Yr2 U
Fall Yr2 U
Spr Yr1 Cert
Spr Yr1 Uncert
0
Fall Yr1 Uncert
1
Decision Stage and Year, Under Uncertainty or Certainty
Figure 6.5. Cropping impacts of water supply certainty for scenario [Full Full Full Full Full Full]. A stage-bystage comparison of activities under uncertainty and certainty. Crop Key: F = furrow, RF = reuse furrow, D =
drip.
139
140
The next scenario analyzed is [Dry Dry Dry Dry Dry Dry]. A producer
who knows in the fall of year 1 that the next six years will be dry is able to change
the crop plan (from the base case crop plan under uncertainty), such that one
additional field of wheat is fully irrigated, and two additional fields of wheat are
substituted for one field each of sugar beet and fallow (table 6.3). Discounted
profit for this scenario under certainty exceeds that under uncertainty by $109,824
(a 27% increase). However, six years of perfectly anticipated drought still reduces
profit by $89,173 (15%), compared to six years of perfectly anticipated full water
allotments. This is a drastic improvement, however, over the profit impact of six
years of unanticipated drought, which reduces profit by $174,430 (30%).
Compared to the uncertainty case, additional fields are planted in this
scenario under certainty. However, the large disparity in profit under certainty
versus uncertainty is largely attributable to the number of attempted versus
successful fields under certainty versus uncertainty (table 6.3). Under uncertainty,
only five of seventeen attempted sugar beet fields are successful, and twenty-five
of twenty-six wheat fields are successful, with one field severely deficit irrigated
(table 6.3). In contrast, under certainty, all four attempted sugar beet fields are
successful, and all twenty-seven attempted wheat fields are successful with less
severe deficit irrigation. No fields are abandoned under certainty. The producer
knows in the fall that the upcoming growing season will be dry, and therefore
attempts exactly the quantity that can be supported.
The retiming of crops does not play a role in the adjusted crop plan for six
dry years, as it did for six full years. All fields in the former scenario are planted
to onions within the first two years of the planning horizon (figure 6.6), even
though the producer knows in advance that six years of drought will occur. This
provides additional evidence that diversification within a year is not an optimal
drought preparedness strategy. The lack of diversification in scenario [Dry Dry
Dry Dry Dry Dry] is because the annual water allotment is sufficiently small that
141
Table 6.3. Number of fields attempted and successfully planted to various
crops in scenario [Dry Dry Dry Dry Dry Dry] under certainty and
uncertainty.
# Fields
Attempted
Crop
Irrig Deficit Cert Uncert
Onion Drip
D1
10
10
Sugbeet Furrow
D1
4
17
Wheat Furrow
D1
12
15
Wheat Furrow
D2
2
0
Wheat Reusefrw
D1
13
11
41
53
# Successful
Fields
Crop
Irrig Deficit Cert Uncert
Onion Drip
D1
10
10
Sugbeet Furrow
D1
4
5
Wheat Furrow
D1
12
11
Wheat Furrow
D2
2
2
Wheat Furrow
D6
0
1
Wheat Reusefrw
D1
13
11
Fallow Furrow
D7
19
20
60
60
142
Onion D
Wheat F
Wheat F 0.5
Wheat F 0.9
Wheat RF
Sg-beet F
Open
Fallow
Crops Chosen for Each of 10 Fields
(each field = 35 acres)
10
9
8
7
6
5
4
3
2
Spr Yr6 C
Spr Yr6 U
Fall Yr6 U
Spr Yr5 C
Spr Yr5 U
Fall Yr5 U
Spr Yr4 C
Spr Yr4 U
Fall Yr4 U
Spr Yr3 C
Spr Yr3 U
Fall Yr3 U
Spr Yr2 C
Spr Yr2 U
Fall Yr2 U
Spr Yr1 Cert
Spr Yr1 Uncert
0
Fall Yr1 Uncert
1
Decision Stage and Year, Under Uncertainty or Certainty
Figure 6.6. Cropping impacts of water supply certainty for scenario [Dry Dry Dry Dry Dry Dry]. A stage-bystage comparison of activities under uncertainty and certainty. Crop Key: F = furrow, RF = reuse furrow, D =
drip, 0.5 = 50% of crop’s irrigation requirement is provided, 0.9 = 90% provided.
142
143
water conserved by the retiming of onions would be insufficient to support
additional fields (table 6.4). The benefit of spreading onions across the planning
horizon is therefore smaller than the opportunity cost. In summary, the primary
effect of unanticipated versus anticipated drought is that more fall-prepared fields
are abandoned; particularly, those prepared for water-intensive sugar beets. This
result is relevant to the discussion of the impact of climate change in section 6.5,
and to the discussion of participation in the multi-peril crop insurance program in
section 6.6.
Table 6.4. Excess water (i.e. the quantity of water remaining after crop
water requirements are met) by year for scenario [Dry Dry Dry Dry Dry
Dry] under uncertainty and certainty.
Uncertainty
Certainty
Excess Water (acre-inches)
Yr1 Yr2 Yr3 Yr4 Yr5 Yr6 Total
12 137 141 313 313 841 1,757
622 300 106 134 106 134 1,401
One final observation regarding drought under certainty versus uncertainty
is that the expected value of a perfect six-year water supply forecast can be
calculated by solving the deterministic model for all sixty-four water supply
scenarios, taking the difference between profit under certainty and uncertainty, and
weighting those differences by the probability of each scenario. There is no
anticipated ability to predict water supplies that far in advance, so this exercise
was not undertaken. The value of a perfect fall forecast for only the upcoming
spring’s water supply is more useful, but also more difficult to estimate. The
model would have to be modified to allow for perfect information for the current
year, but only probabilistic information for the remaining years of the planning
horizon. Due to time constraints, this was not pursued. However, there is no
shortage of literature providing estimates of the value of water supply forecasts
(Adams et al. 1995; Adams et al. 2003; Johnson and Holt 1997; Mjelde and
144
Cochran 1988; Mjelde, Hill, and Griffiths 1998; Mjelde et al. 1988; Solow et al.
1998; Wyse 2004).
6.3
The Role of Inter-year Dynamics
Inter-year dynamics are an important characteristic of many farm systems,
including row crop farms in the study area; yet, they receive limited attention in
the economics literature (Antle 1983). The effects of inter-year dynamics on
drought preparedness and response and the profit impacts of drought are relatively
unexplored. Equations 14 through 21, and 27 through 29 of the binary variables
model (section 4.3) represent agronomic constraints that connect current cropping
decisions to those in past years. Producers indicate that such inter-year dynamics
sometimes result in the persistence of drought’s effects, well after the drought
subsides. The base case solution is therefore examined for evidence of this
persistence.
The set of inter-year agronomic constraints influence the optimal solution
in two ways. They necessitate careful sequencing of crops across space and time,
to maximize profit, which is partly a function of the number of fields of crops that
can be grown over the planning horizon. If crops are not carefully sequenced,
fallow has to be used to break up infeasible crop sequences (e.g. wheat-fallowwheat), and profit opportunities are lost. Inter-year dynamics also require careful
sequencing of crops to manage total water requirements in each year. Suppose
that the sequence of crops that maximizes the number of profitable fields has total
crop water requirements that can only be met with the use of highly efficient (and
thus more costly) irrigation technologies. It might be more profitable to reduce the
total number of fields planted over the planning horizon, such that the total crop
water requirement is reduced and less costly irrigation technologies can be used.
Inter-year crop dynamics, in summary, require the producer to consider the impact
of current decisions on future opportunities.
145
In the presence of inter-year crop dynamics, drought and the producer’s
drought response can generate impacts not only in the years in which the drought
occurs, but in subsequent years as well. Clawson et al. (1980) expressed the
following:
Much of the discussion about the economics of drought
management seems to be concerned, often implicitly, with what to
do when drought strikes and when it continues. It seems
sometimes to be assumed, when the rains finally come, that the
drought has ended and that all is well. The need for help may still
exist, even when the drought is over in an agricultural sense.
Moreover, the form of the recovery from one drought may greatly
affect the flexibility of the persons to deal with the inevitable next
drought.
If this is true, then studies that ignore inter-year dynamics, or limit the
timeframe of their analyses to the years in which drought occurs are likely to
misunderstand drought’s full impact, or worse, to recommend drought
preparedness and response plans that fail to consider what the producer will do
after the drought is over. The base case solution provides an opportunity to ask
whether drought and drought response, in the presence of inter-year dynamics,
generates effects in subsequent years.
6.3.1 Single-Year Drought
The following water supply scenarios are compared first to determine
whether a year 2 drought affects cropping activities and profit after the drought
subsides: (a) [Full Dry Full Full Full Full] and (b) [Full Full Full Full Full Full].
Response to a year 2 drought includes fallowing two fields that were prepared in
the fall for sugar beets, and deficit irrigating two wheat fields (figure 6.7). Net
revenue in year 2 is $25,641 less than if no drought occurs (table 6.5). Changes in
both total revenue and total cost occur. Total revenue decreases because sugar
beets that are not planted cannot be sold, and because yield in the deficit irrigated
146
wheat fields is less than if fully irrigated. Total cost decreases because spring
planting costs for sugar beets are not incurred. The impact of drought in the year
in which it occurs is straightforward. Considered next is whether inter-year crop
dynamics generate impacts in years subsequent to the drought.
Differences in the two scenarios’ profit and cropping activities subsequent
to the drought would indicate that the impacts of drought are not isolated to the
year in which it occurs. The two scenarios’ profits in years subsequent to the
drought are, in fact, not equal (table 6.5). One might expect profit following the
drought to be lower in scenario (a) than in scenario (b) because less profit in year 2
implies less earned interest in subsequent years. Scenario (a)’s profit is indeed
lower in year 6, and the subsequent six years, which capture terminal values.
Profit in years 3 through 5, however, is higher for scenario (a). This is because
drought affects profit in subsequent years not just through reductions in earned
interest, but also through changes in cropping activities.
The two scenarios’ cropping activities differ in years subsequent to the
drought (figures 6.7 and 6.8). Drought’s role in these differences is clear for years
3 and 4. The producer, having abandoned two sugar beet fields during the year 2
drought, reattempts those fields in subsequent years. Specifically, sugar beet
production is increased from three to four fields in both years 3 and 4 (note:
scenario (b)’s activities serve as the reference point). This requires them, however,
to adjust other cropping activities as well, because water is insufficient, ceteris
paribus, to support an extra field of sugar beets. Adjustments include removing
grain corn from the crop plan in years 3 and 4 to accommodate sugar beets, and
using reuse furrow on one additional field of wheat. The above adjustments to the
crop plan in response to drought result in higher profit in years 3 and 4 than in
scenario (b) (table 6.5). Profit lost in year 2 is therefore partially recaptured in
years 3 and 4. An economic analysis that focuses on activities during the year of
drought alone will fail to capture this post-drought rebound.
147
Onion D
Wheat F
Wheat RF
Wheat F 0.9
Sg-beet F
Open
Gcorn F
Gcorn RF
Fallow
9
8
7
6
5
4
3
2
Fall Yr6, {FFFFF}
Fall Yr6, {FDFFF}
Spr Yr6, {FFFFF} Full
Spr Yr6, {FDFFF} Full
Fall Yr5, {FFFF}
Fall Yr5, {FDFF}
Spr Yr5, {FFFF} Full
Spr Yr5, {FDFF} Full
Fall Yr4, {FFF}
Fall Yr4, {FDF}
Spr Yr4, {FFF} Full
Spr Yr4, {FDF} Full
Fall Yr3, {FF}
Fall Yr3, {FD}
Spr Yr3, {FF} Full
Spr Yr3, {FD} Full
0
Fall Yr2, {F}
Spr Yr2, {F} Full
Spr Yr2, {F} Dry
1
Fall Yr1
Spr Yr1, Full
Crops Chosen for Each of 10 Fields
(each field = 35 acres)
10
Decision Stage and Year, {Past} Current Water Supply
Figure 6.7. Cropping impacts of a year 2 drought. A stage-by-stage comparison of optimal cropping activities
for scenarios [Full Full Full Full Full Full] and [Full Dry Full Full Full Full] of the base case. Crop Key: F =
furrow, RF = reuse furrow, D = drip, 0.9 = 90% of crop’s irrigation requirement is provided.
147
148
Table 6.5. Impact of a year 2 drought on undiscounted profit. A year-by-year comparison of undiscounted
profit for scenarios (a) [Full Dry Full Full Full Full] and (b) [Full Full Full Full Full Full] of the base case
optimal solution.
Undiscounted Profit ($)
6-Year Period of Active Production
6-Year Period of Terminal Values
Scenario Yr1
Yr2
Yr3
Yr4
Yr5
Yr6
Yr7
Yr8
Yr9
(a)
74,121 26,257 32,727 35,018 22,946 23,054 160,052 108,683 69,362
(b)
74,121 51,898 30,180 32,293 19,885 25,764 161,453 110,183 70,966
(a)-(b)
0 -25,641 2,547 2,725 3,061 -2,710 -1,401 -1,499 -1,604
Yr10
74,217
75,934
-1,717
Yr11
79,413
81,249
-1,837
Yr12 Total
84,972 790,823
86,937 820,863
-1,965 -30,040
Table 6.6. Impact of a year 3 drought on undiscounted profit. A year-by-year comparison of undiscounted
profit for scenarios (b) [Full Full Full Full Full Full] and (c) [Full Full Dry Full Full Full] of the base case
optimal solution.
Undiscounted Profit ($)
6-Year Period of Active Production
6-Year Period of Terminal Values
Scenario Yr1
Yr2
Yr3
Yr4
Yr5
Yr6
Yr7
Yr8
(c)
74,121 51,898 6,820 25,102 32,415 28,844 160,407 109,064
(b)
74,121 51,898 30,180 32,293 19,885 25,764 161,453 110,183
(c)-(b)
0
0
-23,361 -7,191 12,530 3,080 -1,046 -1,119
Yr9
69,769
70,966
-1,198
Yr10
74,653
75,934
-1,281
Yr11
79,878
81,249
-1,371
Yr12
85,470
86,937
-1,467
Total
798,439
820,863
-22,424
148
149
(i)
(ii)
Year 1 [Full]
Year 1 [Full]
O
O
SB
F
W
O
O
SB
F
W
O
W
O
O
O
O
W
O
O
O
Year 2 [Dry]
Year 2 [Full]
F
W
O
SB
O
F
W
O
F
O
W
O
SB
W
W
W
O
F
W
W
Year 3 [Full]
Year 3 [Full]
W
GC
W
W
W
W
SB
W
W
SB
SB
W
F
SB
SB
SB
W
W
F
SB
Year 4 [Full]
Year 4 [Full]
SB
W
GC
F
SB
SB
W
F
SB
W
W
SB
W
W
W
W
SB
SB
W
W
Year 5 [Full]
Year 5 [Full]
W
SB
W
W
GC
W
F
W
W
GC
F
W
GC
GC
GC
GC
W
W
SB
GC
Year 6 [Full]
Year 6 [Full]
GC
W
SB
F
W
GC
W
SB
GC
W
W
GC
W
GC
W
W
GC
F
W
W
Figure 6.8. Crops assigned to each field (first row of each box reads
from left to right, F1 to F5; second row reads from left to right, F6 to
F10) in each year of the six-year planning horizon. Boxes on the left
(column (i)) are for scenario (b) [Full Full Full Full Full Full]; boxes on
the right (column (ii)) are for scenario (a) [Full Dry Full Full Full Full].
Key: O=onion, SB=sugar beet, W=wheat, GC=grain corn, F=fallow.
Bold letters in column (ii) indicate fields whose crops differ from those
in column (i).
150
The producer’s ability to rebound from drought by re-planting an
abandoned crop is possible because of the agronomic constraints, which prevent
producers from continuously growing an individual crop. Suppose instead that the
producer could continuously plant an individual crop without risk of pest, disease,
or depleted soil quality. If the producer was forced to abandon the crop in year t,
they could not replant in subsequent years without displacing the current year’s
crop. The producer in the base case model, in contrast, can replant an abandoned
field of sugar beets, for example, without displacing next year’s sugar beet crop
because the field is eligible for sugar beets only once every five years. Future
sugar beet crops in that field are delayed; however, the farm’s total sugar beet
acreage over the six-year crop plan is not reduced.
Although some profit is recovered after the drought by replanting crops in
later years, the portion that is not recovered causes reductions in earned interest,
and these reductions compound through time. Eleven years after the drought,
losses attributable to drought total $30,040, a 17% increase from their initial value
of $25,641. This contrasts to losses just five years after the drought, which total
$20,018, a 22% decrease from profit loss during the drought. Reductions in
earned interest more than offset the preliminary recovery of losses via replanting.
This result changes, however, when discounting is considered. The total loss of
discounted profit eleven years after the drought is $24,700, which is very close to
the initial loss of $25,641. Nonetheless, studies that only examine drought’s
immediate crop and profit impacts might misestimate total impacts and
misinterpret subsequent crop choices.
There are additional differences in the crop plans of scenario (a) and (b).
However, not all differences are caused by the year 2 drought. Specifically, crops
in fields F5, F8 and F9 of year 3 are different in the two scenarios (figure 6.8), but
not because of drought. The crops allocated to these fields in column (ii) allow the
producer to grow an additional successful field of wheat in year 5. However, the
151
same crop plan for these three fields could also be followed in scenario (b). The
approximate optimal solution simply did not detect the opportunity to increase
undiscounted profit by $4,500. This implies that undiscounted profit in year 5
should be similar for the two scenarios (table 6.5), and total profit loss attributable
to drought is actually higher than $30,040.
It is more difficult to determine whether the drought in year 2 causes the
differences in cropping activities for years 5 and 6. Close inspection of figure 6.8
reveals, however, that some of the differences in year 5 activities can be traced
back to the drought. In year 5, sugar beets are planted in field F9 in column (ii),
rather than in F2 as in column (i). This has no effect on profit in year 5, but
illustrates that drought can generate impacts several years after it subsides. Sugar
beets’ location on the farm is altered in year 5 because F9 is the only field that
remains eligible. Field F2, which was the planned location prior to the drought,
was planted to sugar beets in year 3, in an effort to recover one of the fields of
beets abandoned in year 2. The remaining crop differences in year 5 and 6,
however, including the extra field of wheat in year 5 of column (ii), are
attributable to the approximate nature of the optimal solution, as discussed above.
Sufficient evidence exists to conclude that drought in year 2, or more precisely, the
producer’s response to the drought, generates impacts not only during the year in
which drought occurs, but also in subsequent years.
6.3.2 Multi-Year Drought
The inter-year dynamics of the model also enable the study of multi-year
or prolonged drought, a topic that has received relatively little attention in the
literature (Iglesias, Garrido, and Gomez-Ramos 2003; Tapp et al. 1998; Thompson
et al. 1996; Toft and O'Hanlon 1979; Ward et al. 2001). The potential for the
impacts of one year of drought to modify the impacts of a subsequent year of
drought is of particular interest (Clawson et al. 1980). Consecutive years of
drought have the potential, because of inter-year crop dynamics, to generate
152
complex impacts on cropping activities and profit. A comparison of the following
four scenarios is made to understand the potential impacts of a two-year drought
occurring in years 2 and 3: (a) [Full Dry Full Full Full Full], (b) [Full Full Full Full
Full Full], (c) [Full Full Dry Full Full Full], and (d) [Full Dry Dry Full Full Full].
A year 2 drought generates a loss of $30,040 in undiscounted profit (table
6.5). A year 3 drought generates a loss of $22,424 (table 6.6). If the impacts of
these droughts were isolated within the year in which they occurred, two outcomes
would be expected: 1) the profit impact of a two-year drought that occurs in years
2 and 3 should be approximately equal to the sum of the individual droughts’
impacts ($52,464), and 2) the losses attributable to a year 3 drought should be the
same regardless of whether it preceded by a dry or full year. The result, however,
is that a two year drought generates a loss of $85,737 (table 6.7), which is much
larger than the hypothesized loss of $52,464. Also, the impact of a year 3 drought
is $55,697 when preceded by a year 2 drought (table 6.8), and only $22,424 when
not preceded by drought (table 6.6). These two results indicate that the impact of a
year 3 drought depends on whether it was preceded by drought, or equivalently,
that the impact of a year 2 drought depends on whether a drought is revealed in
year 3. The results suggest, more generally, that the impact of a multi-year
drought is more complex than the sum of its parts. This result also reinforces the
previous subsection’s conclusion, i.e. that the impact of drought in a farm system
that has inter-year dynamics can continue after the drought subsides.
The impact of a year 3 drought is larger when preceded by a year 2 drought
than when not because of the response to the year 2 drought. Specifically, the
producer attempts to recover from the year 2 drought by preparing four fields for
sugar beets in the fall of year 3, rather than three fields (figure 6.7). When drought
is revealed in the spring of year 3, the producer has to abandon three fields, rather
than two (figure 6.9). Investments in fall field preparation are sunk, so the
producer receives no return on a fall-prepared field that is later abandoned.
153
Table 6.7. Impact of a two-year drought (years 2 and 3) on undiscounted profit. A year-by-year comparison of
undiscounted profit for scenarios (b) [Full Full Full Full Full Full] and (d) [Full Dry Dry Full Full Full] of the
base case optimal solution.
Undiscounted Profit ($)
Period of Active Production
Period to Capture Terminal Values
Scenario Yr1
Yr2
Yr3
Yr4
Yr5
Yr6
Yr7
Yr8
(d)
74,121 26,257 -2,091 26,596 30,101 22,026 157,454 105,904
(b)
74,121 51,898 30,180 32,293 19,885 25,764 161,453 110,183
(d)-(b)
0
-25,641 -32,271 -5,697 10,216 -3,738 -3,999 -4,279
Yr9
66,388
70,966
-4,579
Yr10
71,035
75,934
-4,899
Yr11
76,007
81,249
-5,242
Yr12
81,328
86,937
-5,609
Total
735,126
820,863
-85,737
Table 6.8. Impact on undiscounted profit of a year 3 drought when preceded by a year 2 drought. A year-byyear comparison of undiscounted profit for scenarios (a) [Full Dry Full Full Full Full] and (d) [Full Dry Dry Full
Full Full] of the base case optimal solution.
Undiscounted Profit ($)
Period of Active Production
Period to Capture Terminal Values
Scenario Yr1
Yr2
Yr3
(d)
74,121 26,257 -2,091
(a)
74,121 26,257 32,727
(d)-(a)
-34,818
0
0
Yr4
Yr5
Yr6
Yr7
Yr8
26,596 30,101 22,026 157,454105,904
35,018 22,946 23,054 160,052108,683
-8,422 7,155 -1,028 -2,598 -2,780
Yr9
66,388
69,362
-2,974
Yr10
71,035
74,217
-3,183
Yr11
76,007
79,413
-3,405
Yr12
81,328
84,972
-3,644
Total
735,126
790,823
-55,697
153
154
Onion D
Sg-beet F
Wheat F
Open
Wheat RF
Gcorn F
Wheat F 0.9
Gcorn RF
Wheat RF 0.9
Fallow
9
8
7
6
5
4
3
2
Decision Stage and Year, {Past} Current Water Supply
Fall Yr6, {FFDFF}
Fall Yr6, {FDDFF}
Spr Yr6, {FFDFF} Full
Spr Yr6, {FDDFF} Full
Fall Yr5, {FFDF}
Fall Yr5, {FDDF}
Spr Yr5, {FFDF} Full
Spr Yr5, {FDDF} Full
Fall Yr4, {FFD}
Fall Yr4, {FDD}
Spr Yr4, {FFD} Full
Spr Yr4, {FDD} Full
Fall Yr3, {FF}
Fall Yr3, {FD}
Spr Yr3, {FF} Dry
Spr Yr3, {FD} Dry
0
Fall Yr2, {F}
Spr Yr2, {F} Full
Spr Yr2, {F} Dry
1
Fall Yr1
Spr Yr1, Full
Crops Chosen for Each of 10 Fields
(each field = 35 acres)
10
Figure 6.9. Cropping impacts of a year 3 drought when preceded by a full versus dry year 2. A stage-by-stage
comparison of activities for scenarios [Full Full Dry Full Full Full] and [Full Dry Dry Full Full Full] of the base
case. Crop Key: F = furrow, RF = reuse furrow, D = drip, 0.9 = 90% of crop’s irrigation requirement is
provided.
154
155
Fifteen sugar beet fields are attempted in scenario (d), but only ten are successful.
This contrasts to eleven attempts and successes in scenario (b). Similarly, only
five fields are planted to grain corn in scenario (d) versus nine in scenario (b).
Following the second year of drought, the producer’s approach to recovery
is the same. They attempt four sugar beet fields in year 4, and thus face the same
risk of abandoning more fields of sugar beets in the event of a dry spring. The
producer, in scenario [Full Dry Dry Dry Full Full], would attempt sugar beets in
eighteen fields, with only nine successes. In scenario [Full Dry Dry Dry Dry Dry],
sugar beets would be attempted in eighteen fields, with only seven successes.
Only one field of grain corn is grown in each of these scenarios, in contrast to nine
in scenario (b). The cropping and profit impacts of an unanticipated multiyeardrought are complex; specifically, they are more than just a scaled version of a
single-year drought’s impacts.
The above results have important implications for government assistance in
the event of drought. Suppose a government official asks a producer, after
enduring one year of drought, to report profit impacts of the drought. A producer
in the study area should answer, “I don’t know yet.” The total impact of a drought
will depend on water supplies in subsequent years. The producer will initially
recover some of their loss if they receive a full allotment next year. In contrast,
their loss will be larger if next year is also dry.
6.4
The Role of Crop History
The base case solution for the binary variables model assumes no crop
history prior to the first year of production. Fields are assumed to have had no
crops on them in the previous six years that would limit the producer’s planting
options in the next six years. The producer can therefore consider any crop plan as
long as it meets the agronomic constraints defined for years 1 through 6.
Producers, in reality, have a crop history, one that potentially limits their ability to
follow the base case optimal solution. It is therefore useful to consider the degree
156
to which the optimal drought preparedness and response tools identified in section
5.1 can be transferred to farms that have crop history. An infinite number of crop
histories are possible in reality; this section illustrates the impact of only one
example crop history on the producer’s optimal drought preparedness and response
plan. This example is hereafter referred to as the “history” model.
The crop history used in this example (table 6.9) approximates the crop
history of a row crop producer in the study area during the period 1997 to 2002.
Crops that do not appear in the base case optimal solution appear in the crop
history, including potatoes, alfalfa and silage corn. The producer, after whom the
history was modeled, has a livestock enterprise in addition to the crop enterprise.
Some of the crops are used as an input to the livestock enterprise, which might
justify their appearance in the crop history, but not in the base case solution.
Potatoes are not an input to livestock in this case, but a favorable contract with the
local processor could have caused them to enter the producer’s plan.
Table 6.9. Crop history used in the “history” model. Year 1 of the historic
period occurred six years prior to the current planning period. Year 6 of
the historic period occurred one year prior to the current planning period.
Crop
Onion
Winter Wheat
Russet Potato
Sugar Beet
Alfalfa-1 yr old
Alfalfa-2 yrs old
Alfalfa-3 yrs old
Alfalfa-4 yrs old
Grain Corn
Silage Corn
Total
Year in the Historic Period
1
2
3
4
5
6
(# of fields)
1
2
2
2
1
2
3
2
2
2
2
1
1
1
1
1
1
1
2
2
1
1
2
3
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
10 10 10 10 10 10
157
The resulting optimal crop plan (figures 6.10 through 6.12) is similar to
that of the base case. Specifically, fields are left open in the fall; reuse furrow is
used in some fields; open fields and sugar beets are fallowed in dry years; wheat is
deficit irrigated in dry years, and grain corn is planted in some full years, if water
is available. Crop history alters the optimal crop plan, however, in several
important ways. First, alfalfa appears in years 1 and 2, while it does not appear in
the base case solution. This is because young alfalfa stands appear in year 6 of the
crop history, and the producer is required to keep alfalfa, once planted, in place for
four years. This constraint enforces the use of alfalfa in the study area for longterm maintenance of soil quality. The model is unable to systematically capture
the agronomic benefits of alfalfa in soil quality maintenance, which combined with
its high water requirements, prevents alfalfa from entering the base case solution.
Crops Chosen for Each of 10 Fields
(each field = 35 acres)
Wheat F
Wheat RF
Alf3 RF
Alf4 RF
Sg-beet F
Open
Fallow
10
9
8
7
6
5
4
3
2
1
0
Fall
Spr, Full
Spr, Dry
Decision Stage, Current Water Supply
Figure 6.10. Optimal fall and spring activities in year 1 of the “history”
case. Crop Key: F = furrow, RF = reuse furrow.
Crops Chosen for Each of 10 Fields
(each field = 35 acres)
158
10
Onion D
Alf4 WL 0.6
Wheat F
Alf4 WL 0.9
Wheat RF
Open
Alf4 RF
Fallow
Alf4 WL
Gcorn F
9
8
7
6
5
4
3
2
1
0
Fall, {F} Spr, {F} Spr , {F}
Full
Dry
Fall, {D} Spr, {D} Spr, {D}
Full
Dry
Decision Stage, {Past} Current Water Supply
Figure 6.11. Optimal fall and spring activities in year 2 of the “history”
case. Crop Key: F = furrow, WL = wheel line, RF = reuse furrow, D =
drip, 0.9 = 90% of crop’s irrigation requirement is provided, 0.6 = 60%
provided.
Spr, {DD} Dry
Spr, {DD} Full
Wheat F 0.5
Fall, {DD}
Spr, {DF} Dry
Wheat F 0.6
Fallow
Spr, {DF} Full
Fall, {DF}
Wheat F 0.9
Gcorn F
Spr, {FD} Dry
Spr, {FD} Full
Wheat F
Open
Fall, {FD}
Spr, {FF} Dry
10
9
8
7
6
5
4
3
2
1
0
Spr, {FF} Full
Onion D
Sg-beet F
Fall, {FF}
Crops Chosen for Each of 10 Fields
(each field = 35 acres)
159
Decision Stage, {Past} Current Water Supply
Figure 6.12. Optimal fall and spring activities in year 3 of the “history”
case. Crop Key: F = furrow, D = drip, 0.5 = 50% of crop’s irrigation
requirement is provided, 0.6 = 60% provided, 0.9 = 90% provided.
Alfalfa is also deficit irrigated in year 2, rather than wheat, for the same
reasons that sugar beets are deficit irrigated, rather than wheat, in the base case. A
related difference is that fewer fields of sugar beets are planted overall in the
history model, because alfalfa displaces sugar beets in years 1 and 2. A third
difference is that onions are planted throughout years 2 through 6, rather than in
years 1 and 2, as in the base case solution. Onions are planted in the history model
as soon as fields become eligible, but the crop history delays this for most fields.
The cost of this delay is foregone earned interest on savings, and higher
discounting of profit because it is delayed. Crop history clearly limits the
producer’s feasible set of crop plans. However, it does not change the
applicability of most drought preparedness and response tools.
160
Crop history limits the feasible set of activities, and should therefore
reduce expected profit, and increase the average impact of drought. Expected
profit in the history case is $55,000 (table 6.10), or 11% less than in the base case
(table 5.2). Maximum profit is $50,000 (9%) less, and minimum profit is $40,000
(10%) less. The average profit impact of drought is $10,000 more in the history
case (figure 6.13 versus 5.6) for all drought categories, except six years, which is
$13,000 less. That is, the effect of drought in the crop history case is 88% worse
for a one-year drought, 32% worse for two years of drought; 26% worse for three
years of drought, 13% worse for four years of drought, 6% worse for five years of
drought, and 7% better for six years of drought. In conclusion, crop history, in this
example, necessitates an alternative solution, one that is less profitable in
expectation and less successful during all but the most extreme drought. Future
modeling efforts should attempt to expand the model’s time horizon, such that one
could test whether a producer with any starting crop history will eventually
transition to the base case solution’s crop plan.
Table 6.10. Summary statistics of the “history” case’s profit outcome.
Statistic
Value ($)
Expected Stream of Discounted Profit
475,036
Standard Deviation of Expected Stream
37,431
Maximum Discounted Profit
538,573
Minimum Discounted Profit
368,101
Change in total discounted profit ($1000)
161
0
-10
-20
-30
-40
-50
-60
-70
-80
-90
-100
-110
-120
-130
-140
-150
-160
-170
-180
1
2
3
4
5
6
Years of Drought
Figure 6.13. Average change in the “history” model’s total discounted
profit (black dashes) by years of drought experienced (as compared to 6
years of full water supply). Gray brackets indicate the maximum and
minimum impact of drought.
6.5
Price Uncertainty
The economics literature has increasingly emphasized the joint effect of
multiple sources of uncertainty on farm decisions (Isik 2002; Pannell, Malcolm,
and Kingwell 2000; Thompson and Powell 1998). Producers in the study area face
both water supply and output price uncertainty. Onions, which generate much of
the profit in the model, have an especially volatile output price (table 6.11). The
purpose of this section is to determine whether onion price uncertainty changes the
optimal drought preparedness and response plan. Specialization in onions in the
first year of the base case solution is of specific interest, since producers in the
study area tend to diversify instead for reasons that are not yet clear.
162
Table 6.11. Prices received (2004$) by growers for yellow onions over the
period 1995-2004 (Malheur County Extension Service 2004a).
Price Received*
($/cwt)
1995
4.09
1996
6.57
1997
8.50
1998
8.03
1999
2.37
2000
13.01
2001
4.85
2002
5.12
2003
5.51
2004
2.79
6.08
Mean
3.15
Std Dev
Year
*
Prices do not include packing and shipping premiums.
The base case model is modified to accommodate onion price uncertainty
(section 4.5.2). Price uncertainty is represented by three price categories [Hi =
$12.25, Med = $6.00, Lo = $2.50], derived from price data for the study area
(Appendix B.3). Note that the price of onions in category “Med” is the same as
the price in the certainty model. Price uncertainty is assumed to be resolved only
after both fall and spring decisions are made. The producer therefore has no
recourse after the price is revealed. The model’s planning horizon has to be
shortened from six years to three to enable the programming software to
accommodate a third stage decision stage. This modification causes some
discrepancies in the base case solutions of the six versus three-year models. It
would therefore be inconsistent to compare the three-year price uncertainty
model’s solution to that of the six-year base case model. A three-year version of
the base case model is constructed to facilitate comparison. The solution to this
truncated model is used as the reference solution, against which the price
uncertainty model’s solution is compared.
163
The three-year base case solution is largely similar to the six-year base case
solution for years 1 through 3 (figure 6.14). The three-year solution recommends
more sugar beets than the six-year solution, specifically in year 3, because it has
fewer years in which to plant them. The three-year model also plants one
additional field of onions in year 1 instead of year 2, which enables it to
accommodate an additional sugar beet field in year 3. With a three-year base case
solution established, the effects of introducing onion price uncertainty can be
determined.
Optimal cropping activities for scenarios [Full Full Full] (price certainty)
and [Full Med Full Med Full] (price uncertainty) are compared first. Overall, price
uncertainty has little effect on the optimal crop plan (figure 6.15). The timing of
onions remains unchanged, despite a 25% chance of receiving only $2.50 per
hundredweight (cwt), which would result in a net loss of $2000 per acre for fully
irrigated onions under drip irrigation. There is also, however, a 50% chance of
receiving $6.00 per cwt, and a 25% chance of receiving $12.25 per cwt, which
would generate net revenue of $320 and $4400 per acre, respectively. Expected
net revenue under price uncertainty for fully irrigated onions under drip irrigation
is therefore $760 per acre. Under price certainty, the producer receives $6.00 per
cwt, for a per acre net revenue of $320. The price uncertainty model is also solved
for the following price categories: Hi = $9.55, Med = $6.00, Lo = $2.44. These
prices are such that the expected net revenue per acre for onions equals the net
revenue per acre for onions in the price certainty model. The timing of onions,
again, remains unchanged.
The optimal crop plan remains essentially unchanged when price
uncertainty is added because the producer cannot influence the expected outcome
of price uncertainty by manipulating their crop plan. That is, the expectation
operator does not act over the decision variables. A simple example best
illustrates this point. Suppose a producer can grow ten fields of onions (and no
164
Onion D
Wheat F
Wheat RF
Sg-beet F
Sg-beet F 0.9
Open
Fallow
Gcorn F
Crops Chosen for Each of 10 Fields
(each field = 35 acres)
10
9
8
7
6
5
4
3
2
(3) Spr Yr3
(6) Spr Yr3
(3) Fall Yr3
(6) Fall Yr3
(3) Spr Yr2
(6) Spr Yr2
(3) Fall Yr2
(6) Fall Yr2
(3-Yr) Spr Yr1
(3-Yr) Fall Yr1
(6-Yr) Fall Yr1
0
(6-Yr) Spr Yr1
1
(Planning Horizon) Decision Stage and Year
Figure 6.14. Optimal cropping activities for years 1 through 3 of scenario [Full Full Full], as generated by a sixyear versus three-year version of the base case model. Crop Key: F = furrow, RF = reuse furrow, D = drip, 0.9 =
90% of crop’s irrigation requirement is provided.
164
165
Crops Chosen for Each of 10 Fields
(each field = 35 acres)
Onion D
Wheat F
Wheat RF
Sg-beet F
Open
Fallow
Gcorn F
10
9
8
7
6
5
4
3
2
(U) Spr Yr3
(C) Spr Yr3
(U) Fall Yr3
(C) Fall Yr3
(U) Spr Yr2
(C) Spr Yr2
(U) Fall Yr2
(C) Fall Yr2
(P Uncert) Spr Yr1
(P Uncert) Fall Yr1
(P Cert) Fall Yr1
0
(P Cert) Spr Yr1
1
(Price Uncertain or Certain) Decision Stage and Year
Figure 6.15. Optimal cropping activities for years 1 through 3 of scenarios [Full Full Full] (price certainty) and
[Full Med Full Med Full] (price uncertainty), as generated by a three-year version of the base case model. Crop
Key: F = furrow, RF = reuse furrow, D = drip
165
166
other crops) over a two-year period (assuming a zero discount rate), and that each
field generates one unit of output. Assume the producer faces price uncertainty
only, and that two prices are possible, h and l, with probabilities pr(h) and pr(l),
respectively. Prices are assumed independent between years. The producer
considers two options: a) plant all ten fields in year 1, and risk receiving a low
price for all ten fields, or b) plant five fields in each of years 1 and 2, and hope that
a high price will occur in at least one year. The expected profit of option a) is:
[ pr ( h) ⋅ h ⋅10 + pr (l ) ⋅ l ⋅10] + [ pr ( h) ⋅ h ⋅ 0 + pr (l ) ⋅ l ⋅ 0]
= E ( price) ⋅10 + E ( price) ⋅ 0
= E ( price) ⋅10.
The expected profit of option b) is:
[ pr ( h) ⋅ h ⋅ 5 + pr (l ) ⋅ l ⋅ 5] + [ pr ( h) ⋅ h ⋅ 5 + pr (l ) ⋅ l ⋅ 5]
= E ( price) ⋅ 5 + E ( price) ⋅ 5
= E ( price) ⋅10.
Regardless of how the producer allocates the ten onion fields across time, expected
net revenue is the same. Returning to the more complex model, it is clear why the
optimal solution under water supply uncertainty remains unchanged when price
uncertainty is introduced. Retiming of onions does not change the expected
outcome of price uncertainty.
A parallel example for water supply uncertainty reinforces this point.
Suppose the same producer faces water supply uncertainty only, and that two
water supplies are possible, f and d, with probabilities pr(f) and pr(d), respectively.
The producer again considers the following two options: a) plant all ten fields in
year 1 (and if the spring is dry they will have to abandon two fields), or b) plant
five fields in each of years 1 and 2 (and if the spring is dry they will not have to
abandon any fields). The expected profit of option a) is:
167
[ pr ( f ) ⋅10 + pr ( d ) ⋅ 8] + [ pr ( f ) ⋅ 0 + pr (d ) ⋅ 0]
= [ pr ( f ) ⋅10 + pr ( d ) ⋅ 8].
The expected profit of option b) is:
[ pr ( f ) ⋅ 5 + pr (d ) ⋅ 5] + [ pr ( f ) ⋅ 5 + pr (d ) ⋅ 5]
= [5 + 5] = 10.
In contrast to the price uncertainty example, the expected outcome of these two
options is not equal (unless the water supply is certain, i.e. pr(f) equals one). In
conclusion, the hypothetical producer who maximizes expected profit can do
nothing to mitigate the effects of price uncertainty on expected profit, but can
mitigate the effects of water supply uncertainty.
The structure of the price uncertainty problem, as modeled in this study, is
different than that of the water supply uncertainty problem. The expectations
operator in the price uncertainty problem acts on price independent of the
producer’s decisions. The expectations operator in the water supply uncertainty
problem, in contrast, acts on the random variable through total yield, which is a
function of the producer’s decisions. This distinction is illustrated below.
Price Uncertain : E (Net Revenue) = E ( price ⋅ totyield (decisions ))
= E ( price) ⋅ totyield ( decisions )
Water Supply Uncertain : E (Net Revenue) = E ( price ⋅ totyield (decisions, water ))
= price ⋅ E (totyield ( decisions, water ))
≠ price ⋅ totyield ( decisions, E ( water ))
Although price uncertainty has no effect on a risk-neutral producer’s
optimal preparedness and response plan, it does affect the profit outcome. The
addition of price uncertainty increases expected profit by nearly 50% (from
168
$473,000 to $700,000), due to a 25% chance of receiving $12.25 per cwt for
onions rather than $6.00. Price uncertainty also drastically increases profit
variability (from a standard deviation of $26,000 to $980,000), and creates the
potential for large profit losses (table 6.12). A risk-neutral producer would be
unconcerned by this, but a risk-averse producer would seek an alternative crop
plan that reduces profit variability. Crop diversification within each year,
specifically spreading onions over the planning horizon, reduces the standard
deviation of profit from $980,000 to $650,000. This result suggests that producers
in the study area might spread onions through the planning horizon as a risk-averse
response to price uncertainty.
Table 6.12. Summary statistics of the profit outcome ($) for the following
three-year models: (i) water supply uncertainty and onion price certainty,
and (ii) water supply and onion price uncertainty.
Statistic
(i)
(ii)
Expected Stream of Discounted Profit
472,974
701,280
Standard Deviation of Expected Stream 26,179
981,272
Maximum Discounted Profit
507,876 2,597,705
Minimum Discounted Profit
416,048 -755,400
One final observation about the effect of price uncertainty on the base case
solution is that a low onion price is clearly more devastating than a drought (table
D1). A risk-averse producer would therefore likely focus their efforts on
managing price uncertainty (e.g. planting onions throughout the planning horizon,
or contracting onions in advance), rather than water supply uncertainty.
Thompson and Powell (1998) also conclude that price risk is greater than yield risk
for many, but not all Australian farm systems.
6.6
Prevented Planting Provision of the Multi-peril Crop Insurance Program
The multi-peril crop insurance program includes a prevented planting (PP)
provision for irrigated crops. A PP payment is made when an insured producer
169
provides evidence that as of the final planting date they have no reasonable
expectation of receiving sufficient water to follow good irrigation practices, due to
an insurable cause of loss, such as drought (Risk Management Agency 2003).
This contrasts to a traditional MPCI claim, where a crop was planted, but later
failed due to unanticipated drought. Producers in the study area indicate that the
PP provision is a useful drought preparedness tool, and an insurance agent for
producers in the study area attributes participation in MPCI largely to the PP
provision (Agricultural Producers in the Vale Oregon Irrigation District 2003;
Haight 2004). However, to the author’s knowledge, no economic studies have
examined the prevented planting provision in this role. Existing studies have
focused instead on the provision’s susceptibility to adverse selection and
fraudulent claims (Rejesus, Escalante, and Lovell 2005; Rejesus et al. 2003). The
prevented planting provision’s effectiveness at the farm-level as a drought
preparedness tool is therefore analyzed in this section. No attempt is made,
however, to determine the social efficiency of the prevented planting provision.
The producer in the model has the option to purchase alternative coverage
levels of multi-peril crop insurance (each with a fixed level of PP coverage) for
individual fields of the following fall-planted crops: onions, potatoes, sugar beets,
and wheat. The MPCI coverage level, in the context of PP claims, is the percent of
historical crop yield that the insurance company will reimburse in the event of a
loss. The PP coverage level is the percent of the MPCI indemnity that the
producer will receive in the event of a successful PP claim. Crop insurance
policies are purchased in the fall, before the upcoming growing season’s water
supply is known. If the water supply is revealed dry, the producer then chooses
whether to abandon the crop and receive a prevented planting payment, or to plant
the crop. No payment is received if the crop is abandoned during a year in which a
full water allotment is received, or if the insured crop is planted successfully.
170
Claims for post-planting disasters, such as hail, pests, freeze, or abnormally
high temperatures are not modeled. Crop insurance is also not offered in the
model for spring-planted crops. These would both require a decision third stage in
each year of the model. Solution of the prevented planting model is sufficiently
difficult with only two stages per year. The exclusion of spring sources of crop
loss implies that the producer pays the entire multi-peril crop insurance premium,
but receives prevented planting coverage only. The portion of the premium
attributed to prevented planting coverage is unknown, however, so a conservative
approach is taken. The model, as a result, likely underestimates the adoption of
multi-peril crop insurance with prevented planting provisions.
In the base case solution, the producer frequently abandons sugar beets in
response to drought. The producer might therefore purchase multi-peril crop
insurance with prevented planting provisions for at least some portion of their
sugar beet fields. Theoretically, the producer should fully insure a crop (i.e.
purchase insurance coverage equal to the potential loss) if the premium is
actuarially fair. A premium is actuarially fair if it equals the expected insurance
indemnity (i.e. the expected payment to the producer). An actuarially unfair
premium should cause the producer to underinsure. An actuarially favorable
premium should cause the producer to over-insure, if the insurance company
allows it.
Unsubsidized and subsidized premiums (table 6.13) are estimated using the
Risk Management Agency’s online premium calculator for the year 2004 (the year
to which all other cost and price data are calibrated) (2006). Expected insurance
indemnities are estimated assuming alternative values for the expected probability
of crop abandonment in any given year due to drought (i.e. the probability of an
indemnity occurring) (table 6.13). The true abandonment probabilities are likely
unique to each crop; for example, the base case solution indicates that 68% of
attempted sugar beet is abandoned on average during drought, while only 5% of
171
attempted wheat is abandoned during drought, and 0% of onions are abandoned.
The expected probability of abandonment of sugar beets, wheat, and onions in any
given year, assuming no abandonment during a full year and a 40% chance of
drought, is 27%, 3%, and 0%, respectively. However, the base case solution also
illustrates that the proportion of attempted fields abandoned during drought varies
with the suite of accompanying crops. The proportion of abandoned sugar beets
varies from 33 to 100%; the proportion of abandoned wheat varies from 0 to 20%.
Table 6.13. Unsubsidized and subsidized premiums, prevented planting
payment, and expected indemnity ($ per acre) for alternative combinations
of crop and MPCI coverage level. Parameter assumptions used to estimate
PP payments appear in tables C.1, C.2, C.4, and A.2.
Insured
Crop
MPCI
Cover
-age
Level
Unsub
Premium
Onion
50
72
19
402
161
101
20
4
65
128
42
523
209
131
26
5
75
195
71
603
241
151
30
6
Potato
75
97
35
260
104
65
13
3
Sug Beet
50
20
5
272
109
68
14
3
65
37
12
354
142
89
18
4
75
65
24
408
163
102
20
4
85
118
59
462
185
116
23
5
55
6
2
138
55
35
7
1
65
10
3
163
65
41
8
2
75
15
6
188
75
47
9
2
85
28
14
213
85
53
11
2
Wheat
Sub Prevented
Prem- Planting
Payment*
ium
Expected
Indemnity by
Probability of
Abandonment
40% 25% 5% 1%
Table 6.13 reports expected indemnity for alternative probability of
abandonment, specifically 40%, 25%, 5% and 1%. A probability of
172
abandonment equal to 40% assumes that the crop is always abandoned when
drought is revealed; therefore, a 40% probability of drought implies a 40%
probability of abandonment. It is clear from previous models’ solutions that
drought does not imply abandonment for all crops, because other drought
responses are available (e.g. deficit irrigation, or abandon one crop to provide
water for another). The middle and right columns assume, more conservatively,
that a 40% probability of drought implies a 25, 5, and 1% probability of
abandonment, respectively.
Pair-wise comparisons of premium and expected indemnity (table 6.13)
indicate whether premiums are actuarially fair, favorable, or unfair at these
assumed abandonment probabilities. Table 6.14 reports whether premiums are
fair, favorable, or unfair for alternative probabilities of crop abandonment. All
premiums (subsidized or unsubsidized) are actuarially favorable if the true
probability of abandonment is 40%. All subsidized premiums are also actuarially
favorable if the true probability is 25%. Recall that the expected probability of
sugar beet abandonment indicated by the base case solution is approximately 25%.
All premiums are unfair if the true probability of abandonment is 1%, which is the
case for onions. Lastly, the base case solution indicates that the expected
probability of wheat abandonment, when rounded up, is approximately 5%.
Subsidized premiums for wheat are favorable, except for the highest coverage
level; unsubsidized premiums for wheat are unfair, except for the lowest coverage
level. (table 6.14). These results suggest that under subsidized premiums, sugar
beets should be over-insured, onions under-insured, and wheat over-insured, but
not to the maximum degree possible.
It is not immediately clear in the case of prevented planting coverage how a
producer can over or under-insure, since the level of prevented planting coverage
is fixed for each crop (in this model) (table C.4). A producer who anticipates $600
losses for an acre of abandoned onions, for example, cannot over-insure by simply
173
Table 6.14. Actuarial fairness of unsubsidized and subsidized multi-peril
crop insurance (MPCI) premiums, in 2004, for alternative crops and
coverage levels. Key: fair (+), favorable (++), unfair (-).
Insured
Crop
Onion
Potato
Sug Beet
Wheat
MPCI
Cover
-age
Level
50
65
75
75
50
65
75
85
55
65
75
85
Fairness of
Unsubsidized Premiums
by Probability of
Abandonment*
40% 25% 5%
1%
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
-
Fairness of
Subsidized Premiums by
Probability of
Abandonment*
40% 25% 5%
1%
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
-
*
Fairness is reported for alternative probabilities of crop abandonment (40,
25, 5, and 1%). Premiums are actuarially fair if premium = expected
indemnity; favorable if premium > expected indemnity; unfair if premium
< expected indemnity. Premiums and expected indemnities are reported in
table 6.13.
purchasing $700 of prevented planting coverage. A producer can over-insure only
by purchasing a higher MPCI coverage level. Specifically, if they wish to insure
for $700, they solve the following algebraic problem for X: [550 · X · 0.45 · $3.25
= $700]. The equation’s right hand side is the desired payment in the event of a
prevented planting claim. The left-hand side is the equation used to determine the
PP payment; specifically, it is the product of approved yield, MPCI coverage level,
PP coverage level, and the elected price. The producer, in order to receive a $700
payment in the event of onion abandonment, should elect for a MPCI coverage
level of 87%. The producer could, in reality, achieve the same effect by selecting
174
a higher price election for a particular MPCI coverage level, or purchasing
additional PP coverage, which is not allowed for all crops. These two options are
not considered here because the model becomes too large to be solved.
The equation illustrated above is used in combination with the parameters
in tables C.1, C.3 and C.4, and the fall cost of each crop (table A.3) to provide
rough estimates of the MPCI coverage levels necessary to fully insure a producer’s
abandonment losses (table 6.15). To fully insure sugar beets, for example, a
producer solves the following equation for X: [ 31 · X · 0.45 · $39.00 = $150]. Fall
preparation costs of $150 per acre represent the loss if an acre of sugar beets is
abandoned. The producer should adopt an MPCI coverage level of 28% to fully
insure sugar beets, assuming approved yield of 31 ton, prevented planting
coverage of 45%, and a price election of $39.00 per ton. The producer overinsures (under-insures) if they choose MPCI coverage levels greater (less) than
those reported in table 6.15.
Table 6.15. MPCI coverage levels required to fully insure a producer’s
losses from drought-induced abandonment. Parameter values in tables A.3,
C1, C3, and C4 are assumed.
Insured
Crop
Onion
MPCI Coverage
Level (%)
75
Potato
29
Sugar Beet
28
Wheat
64
The prevented planting model’s solution indicates that the producer enrolls
all sugar beet acreage, in all scenarios, at the 75% coverage level. The producer,
as expected, over-insures sugar beets. The number of wheat fields enrolled varies
from 0 to 3 fields depending on the water supply scenario; however, on average,
only one field of wheat is enrolled (out of 5 attempted fields on average), and at
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the unexpectedly high level of 85%. That the producer insures very little of their
wheat compared to sugar beets reflects that wheat is abandoned relatively
infrequently. The producer over-insures wheat, as expected, but to an unexpected
degree. Recall that the subsidized premium for 85% MPCI coverage is unfair at
the 5% abandonment probability. The producer’s enrollment at this unfair level
suggests that the probability of abandonment for wheat is actually higher than 5%,
or that the model’s approximate solution has not identified the optimal coverage
level. Lastly, one onion field is enrolled at the 50% level in a small number of
scenarios. Onions are never abandoned in the optimal solution, so it is unclear
why a producer would enroll any onion field. The enrollment of an occasional
field might be attributable, however, to the approximate nature of the optimal
solution.
The availability of subsidized PP coverage affects both cropping activities
and profit outcomes. Cropping activities change in the following ways. More
fields of sugar beets are attempted; this is because PP coverage reduces the cost of
abandonment in the event of a dry year. The following two adjustments make it
possible to attempt more sugar beets: 1) shifting onions from year 1 to years 2 and
3 to make more complete use of water supplies, and 2) decreasing the number of
open fields, which subsequently reduces the number of grain corn fields.
Although more fields of sugar beets are attempted, fewer fields are successful in
most scenarios. Fewer successful fields is not necessarily undesirable for the
producer, however, particularly if the prevented planting payment for sugar beets
exceeds actual losses. The producer in this model is allowed to and does overinsure, which implies that the payment exceeds actual losses.
The perverse incentive to plant additional sugar beet acreage is partially
offset, however, by the fact that the producer cannot receive a prevented planting
payment in the event of a full water allotment year. The pursuit of prevented
planting payments must therefore be balanced against the consequences of getting
176
trapped in a full year with too many beet fields. The latter could be problematic
due to crop water requirements (e.g. only 6 sugar beet fields can be fully irrigated
given a full allotment; all other fields would have to be abandoned), and
agronomic constraints (e.g. eligibility of fields for sugar beets is quickly
exhausted, so less profitable grain corn has to be rotated with wheat to avoid
exhausting eligibility for wheat). Sugar beets are limited in most years and
scenarios to three or four fields. Five or six fields do appear in some years;
however, a portion of these fields are typically under reuse furrow, which reduces
crop water requirements.
The profit impacts of subsidized prevented planting coverage are
substantial (figure 6.16). Expected profit is increased by 16% ($85,000);
minimum profit is increased by 41% ($150,000); maximum profit is increased by
15% ($88,000), and standard deviation is decreased by 38%. The PP provision
therefore achieves the goal of most farm programs, to stabilize farm income
(Lewandrowski and Brazee 1992). In addition, scenarios dominated by drought
become more profitable than those that are not (figure 6.17). Six years of drought,
for example, is 17% more profitable than six years of full water allotments (table
C.5). Prevented planting coverage effectively eliminates profit losses attributable
to drought. The actual effects of this program on profit in the study area have not
been quantified. However, producers in the study area are generally positive about
the usefulness of the PP provision as a drought management tool. This analysis
suggests that PP coverage, for sugar beets in particular, is a very effective drought
preparedness tool at the farm-level. Note again, however, that this study makes no
attempt to determine whether the PP provision improves social welfare.
177
1.1
Proportion of water supply scenarios
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
350
400
450
500
550
600
650
700
750
Discounted Profit (thousand $)
Figure 6.16. Cumulative distribution function of discounted profit without prevented planting coverage (solid
line) and with subsidized PP coverage (dashed line).
177
178
Change in total discounted profit ($1000)
120
100
80
60
40
20
0
-20
-40
-60
-80
-100
-120
-140
-160
-180
1
2
3
4
5
6
Years of Drought
Figure 6.17. Average change in total discounted profit by years of drought experienced (as compared to 6 years
of full water supply). Black dashes represent average change without prevented planting coverage. Black
asterisks represent average change with subsidized PP coverage. Gray brackets indicate the maximum and
minimum impact of drought with PP coverage (dashed gray) and without (solid gray).
178
179
The potential role of an unsubsidized PP provision as a drought
management tool is investigated next. The solution to the unsubsidized coverage
model indicates that the producer enrolls most, but not all sugar beet acreage at the
65% coverage level. This contrasts to the subsidized model, in which the producer
enrolled sugar beets at the 75% coverage level. Enrollment of wheat ranges across
scenarios from 0 to 3 fields, and at various coverage levels, including 55, 75, and
85%. Lastly, no onion fields are enrolled in prevented planting coverage. Despite
the absence of subsidies, the producer still over-insures sugar beets, and wheat, in
most cases. However, the number of insured fields of sugar beets decreases
slightly, and the degree of over-insuring declines. The effects of unsubsidized
prevented planting coverage on cropping activities are nearly identical to those of
subsidized coverage. More fields of sugar beets are attempted, with fewer fields
successful in most scenarios. Onions are shifted from year 1 to years 2 and 3. The
number of open fields decreases, and thus so does the number of grain corn fields.
The profit impacts of unsubsidized prevented planting coverage remain
positive, from the perspective of reducing drought impacts, but are less extreme.
Expected profit is increased by 11% (versus 16% with subsidized coverage);
minimum profit is still increased by 41%; maximum profit is only increased by 5%
(versus 15%), and standard deviation is decreased by 72% (versus 38%). The
additional decrease in the standard deviation of profit is attributable to a reduction
in the right-tail levels of profit (figure 6.18) Scenarios dominated by drought are
still more profitable than those that are not, but to a lesser degree (figure 6.19).
Six years of drought, for example, is only 5% (versus 17%) more profitable than
six years of full allotment (table C.6). Unsubsidized prevented planting coverage
effectively eliminates profit losses attributable to drought, but does so while
creating less extreme profit improvements (figure 6.18). The prevented planting
provisions, in conclusion, would remain an effective component in producers’
drought management toolbox even if premiums were not subsidized.
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No PP
Unsubsidized PP
Subsidized PP
1.1
Proportion of water supply scenarios
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
350
400
450
500
550
600
650
700
750
Discounted Profit (thousand $)
Figure 6.18. Cumulative distribution function of discounted profit without prevented planting coverage, with
unsubsidized PP coverage, and with subsidized PP coverage.
180
181
40
Change in total discounted profit ($1000)
20
0
-20
-40
-60
-80
-100
-120
-140
-160
-180
1
2
3
4
5
6
Years of Drought
Figure 6.19. Average change in total discounted profit by years of drought experienced (as compared to 6 years
of full water supply). Black dashes represent average change without prevented planting coverage. Black
asterisks represent average change with unsubsidized PP coverage. Gray brackets indicate the maximum and
minimum impact of drought with PP coverage (dashed gray) and without (solid gray).
181
182
6.7
Climate Change
One anticipated effect of climate change in the west is that mountain
precipitation will be received increasingly as rain, rather than snow (Frederick and
Gleick 1999; Intergovernmental Panel on Climate Change 2001a; Knowles,
Dettinger, and Cayan 2006). Snow pack levels are also expected to form much
later in the winter, to accumulate in much smaller quantities, and to melt earlier in
the season (Intergovernmental Panel on Climate Change 2001a; Stewart, Cayan,
and Dettinger 2004). Observation of the runoff process and resulting reservoir
levels in the study area by the regional water master and water district manager
indicate the potential for climate change to decrease runoff to reservoirs, and
therefore increase the frequency of water shortage (Jacobs 2004; Ward 2004). The
effects of increased drought frequency and severity on drought preparedness and
response, and on the profit impact of drought, are examined in this section.
No estimate has been made yet of the expected increase in the frequency or
severity of dry years in the study area. The following three climate-change
scenarios, which are the focus of this section’s analysis, are therefore hypothetical
in nature: 1) a 25% increase in the probability (frequency) of drought from 40 to
50%, 2) a 25% increase in drought severity from 24 to 18 acre-inches per acre, and
3) a 25% increase in both drought frequency and severity. The three climate
change scenarios’ relative impact on cropping patterns and expected farm profit, as
compared to the base case solution, is discussed next.
Producers adapt to increased drought frequency (case 1) by reducing the
number of fields prepared for sugar beets, a water-intensive crop, and increasing
those prepared for wheat, a less water-intensive crop. Although wheat, like sugar
beet, is subject to the risk of abandonment, it is less prone to abandonment because
it requires less water. Wheat is also more profitable under deficit irrigation than
beets, which creates a viable alternative to abandonment in the event of a drought.
Shifting the crop mix is considered one of the least-costly means of adjusting to
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climate change (Lewandrowski and Brazee 1992; Mjelde et al. 1997). More
efficient irrigation technologies are also used on a larger number of fields;
specifically, wheat is grown primarily under reuse furrow rather than furrow
irrigation (figure 6.20). If the producer continued to implement the base case
solution under increased drought frequency, more deficit irrigation and
abandonment of fall-prepared sugar beet fields would be necessary. The case 1
solution indicates that it is instead more profitable to shift the crop mix by
increasing the proportion of fields prepared for less water-intensive crops, and
using more efficient irrigation technology.
Note that an increase in the probability of drought does not affect the
timing of onions. A sensitivity analysis indicates that the timing of onions across
years is not affected even by very extreme increases in the probability of drought
(e.g. from 40 to 80%). It has already been determined, however, that the water
supply during drought, defined as 24 acre-inches per acre, allows up to seven
fields of drip-irrigated onions to be fully irrigated. Recall that the benefit of
retiming onions is that it provides opportunities to support additional fields over
the planning horizon by balancing total crop water requirements through time.
This benefit is largest when water supplies are full. As the probability of a full
allotment declines, due to climate change, the expected benefit of retiming onions
declines.
Producers adapt to increased drought severity (case 2) by shifting one field
of onions from year 1 to 2. Water is insufficient during a more severe drought to
fully support seven fields, and the fall preparation cost of onions is too high to risk
abandoning them. Producers also shift some wheat production from furrow to
reuse furrow irrigation, which reduces wheat’s net irrigation requirement, and
subsidizes other crops’ water needs, such as sugar beets (figure 6.21). The number
of fields prepared for sugar beets increases under increased drought severity. This
contrasts with the response to increased drought frequency. An increase in
Crops Chosen for Each of 10 Fields
(each field = 35 acres)
0
Sg-beet F
Open
Fallow
[Probability of Drought] Decision Stage and Year, {Past} Current Water Supply
[0.4] Fall Yr6, {FDDFF}
[0.5] Fall Yr6, {FDDFF}
[0.4] Spr Yr6, {FDDFF} Full
[0.5] Spr Yr6, {FDDFF} Full
Wheat RF
[0.4] Fall Yr5, {FDDF}
[0.5] Fall Yr5, {FDDF}
[0.4] Spr Yr5, {FDDF} Full
[0.5] Spr Yr5, {FDDF} Full
Wheat F 0.9
[0.4] Fall Yr4, {FDD}
[0.5] Fall Yr4, {FDD}
[0.4] Spr Yr4, {FDD} Full
[0.5] Spr Yr4, {FDD} Full
Wheat F
[0.4] Fall Yr3, {FD}
[0.5] Fall Yr3, {FD}
[0.4] Spr Yr3, {FD} Dry
[0.5] Spr Yr3, {FD} Dry
Onion D
10
[0.4] Fall Yr2, {F}
[0.5] Fall Yr2, {F}
[0.4] Spr Yr2, {F} Dry
[0.5] Spr Yr2, {F} Dry
[Pr(D)=0.4] Fall Yr1
[Pr(D)=0.5] Fall Yr1
[Pr(D)=0.4] Spr Yr1, Full
[Pr(D)=0.5] Spr Yr1, Full
184
Gcorn F
Gcorn RF
9
8
7
6
5
4
3
2
1
Figure 6.20. Cropping impacts of increased drought frequency (from 40 to 50%) on scenario [Full Dry Dry Full
Full Full]. A stage-by-stage comparison of activities under a probability of drought of 0.4 versus 0.5. Crop Key:
F =furrow, RF = reuse furrow, D = drip, 0.9 = 90% of crop’s irrigation requirement is provided.
184
Crops Chosen for Each of 10 Fields
(each field = 35 acres)
10
9
8
7
6
5
4
3
2
1
0
Wheat F 0.9
Open
Gcorn F
[24] Fall Yr6, {FDDFF}
[18] Fall Yr6, {FDDFF}
[24] Spr Yr6, {FDDFF} Full
[18] Spr Yr6, {FDDFF} Full
Sg-beet F
[24] Fall Yr5, {FDDF}
[18] Fall Yr5, {FDDF}
[24] Spr Yr5, {FDDF} Full
[18] Spr Yr5, {FDDF} Full
Wheat RF
[24] Fall Yr4, {FDD}
[18] Fall Yr4, {FDD}
[24] Spr Yr4, {FDD} Full
[18] Spr Yr4, {FDD} Full
[24] Fall Yr3, {FD}
[18] Fall Yr3, {FD}
[24] Spr Yr3, {FD} Dry
[18] Spr Yr3, {FD} Dry
Wheat F
[24] Fall Yr2, {F}
[18] Fall Yr2, {F}
[24] Spr Yr2, {F} Dry
[18] Spr Yr2, {F} Dry
Onion D
[D=24] Fall Yr1
[D=18] Fall Yr1
[D=24] Spr Yr1, Full
[D=18] Spr Yr1, Full
185
Gcorn RF
Fallow
[Acre-inches per acre in a Dry year] Decision Stage and Year, {Past} Current Water Supply
Figure 6.21. Cropping impacts of increased drought severity (from 24 to 18 acre-inches/acre) on scenario [Full
Dry Dry Full Full Full]. A stage-by-stage comparison of activities under the base case versus more severe
drought. Crop Key: F = furrow, RF = reuse furrow, D = drip, 0.9 = 90% of crop’s irrigation requirement is
provided.
185
186
drought severity will require more fields to be abandoned and fallowed. Failed
sugar beet fields are typically reattempted in later years; increased failure therefore
implies increased attempts over the planning horizon.
It is not immediately clear why wheat is not substituted for sugar beets in
the case of increased drought severity, as it is for the case of increased drought
frequency. Substitution of wheat for sugar beets reduces the abandonment of
fields in the case of increased drought frequency, and so would seem beneficial in
the case of more severe drought as well. It is not, however, because water is
sufficiently short during the more severe drought to require even some wheat
fields to be abandoned. Wheat requires larger fall investments than sugar beets,
and is thus more costly to abandon. Wheat is substituted for sugar beets for its
ability to avoid abandonment, but its ability to do so depends on the severity of the
drought. Additionally, sugar beets, in the event of a full year and full irrigation,
are more profitable than wheat. Again, if drought leads inevitably to field
abandonment, a producer should prepare fields for the crop with a lower fall cost
and higher profit in the event of a full year. Sensitivity analysis confirms the role
of fall costs in the substitution, or lack thereof, of wheat for sugar beets. A
reduction in the fall cost of wheat causes wheat to be substituted for sugar beets in
case 2, just as in case 1.
As the severity of drought worsens, onions are increasingly spread across
multiple years to avoid abandonment of onion fields. If drought is defined as 12
acre-inches per acre, for example, the producer plants four fields to onions in each
of years 1 and 2, followed by two fields in year 3. More modest increases in
drought severity, from 24 to 22 or 20 acre-inches per acre for example, do not
affect the timing of onions, but do prompt a shift from furrow to reuse furrow
irrigation of wheat. The shift of onions across an increasing number of fields as
drought severity increases indicates some drought-preparedness role for the timing
of onions. The tendency of producers in the study area to spread onions through
187
time could reflect, in part, a positive probability of sufficiently severe drought.
This would justify planting only one or two fields of onions per year, which may
be all that can be supported during these severe droughts.
The results above indicate that the response to climate change depends on
which climate characteristics change. The only similarity in response to a change
in drought frequency versus severity is a shift from furrow to reuse furrow
irrigation for wheat production. Reuse furrow irrigation, coincidentally, is
becoming increasingly prominent in the study area. The technology was originally
introduced there to reduce agricultural runoff and improve water quality.
Producers who have adopted this technology are likely, however, to experience
additional benefits from it if the predicted impacts of climate change materialize.
Producers adapt to an increase in both the frequency and severity of
drought (case 3) in much the same manner as they adapt to case 2. Adaptations to
case 3 include the following: a) shifting one field of onions from year 1 to 2, b)
continuing to attempt sugar beets, rather than shifting to wheat, despite more
frequent abandonment, which results in more fallowing in dry years, and c)
replacing furrow irrigation with reuse furrow on many wheat fields. There is also
a unique adaptation to case 3. The number of wheat fields attempted is reduced,
and the number of open fields is increased. This adaptation, in an environment of
more frequent and severe drought, reduces the cost of fallowing in dry years and
increases grain corn production in full years. The producer, in summary, alters
several features of their crop plan to mitigate the impacts of more frequent and
severe drought. The impact of each climate change case on profit is discussed
next.
The impact of increased drought severity (case 2) on expected profit, when
responded to optimally, is nearly three times that of increased drought frequency
(case 1) (table 6.16). Increased drought severity also significantly reduces
minimum profit, which is opposite of increased drought frequency, which actually
188
increases minimum profit. Lastly, the variability of profit increases by more than
half in case 2, whereas variability decreases in case 1. The hypothetical producer
is better able to mitigate for more frequent moderate drought than for less frequent
but more severe drought. Suppose that both characteristics change simultaneously.
The impact of an increase in both drought frequency and severity on expected
profit is 50% larger than that of severity alone. Additionally, profit variability
increases and minimum profit is reduced, as in case 2. Maximum profit also
declines by more in case 3.
Table 6.16. Profit impacts ($1,000) of three climate change cases as
compared to the base case: 1) increase in the frequency of drought from 40
to 50%, 2) increase in drought severity from a 24 acre-inch/acre water
allotment to an 18 acre-inch water allotment, and 3) an increase in drought
frequency and severity. Numbers in brackets indicate percent change from
the base case.
Case
Base
Frequency
(% years)
40
Severity
(ac-inches/ac)
24
E(π)
532
Std Dev
36
Min π
408
Max π
590
1
50
24
515 [-3]
28 [-21]
429 [+5]
572 [-3]
2
40
18
476 [-11]
59 [+65]
272 [-33]
572 [-3]
3
50
18
448 [-16]
55 [+51]
302 [-26]
552 [-7]
The profit-maximizing producer is clearly worse off, in terms of expected
profit, in each of the climate change scenarios examined. An increase in drought
severity, or both severity and frequency leave the producer significantly worse off.
The average profit impact of drought, however, is not necessarily worse for all
climate change scenarios (table 6.17). An increase in the frequency of drought
(case 1) actually decreases the percent profit impact of drought. As the probability
of drought increases, the crop plan is increasingly tailored to drought, rather than
the less common full year; therefore, the difference between expected profit and
profit during drought is less than in the base case solution.
189
An increase in drought severity (case 2), in contrast, nearly doubles the
percent profit impact of all drought categories. That is, the producer’s expected
profit declines, and the average impact of drought worsens. This again reflects the
producer’s difficulty in effectively mitigating more severe water shortages, in
contrast to more frequent but moderate shortages. Lastly, an increase in both
drought frequency and severity (case 3) also causes larger percent profit losses
attributable to drought. The percent losses are slightly less than those experienced
in case 2, but more than those in case 1. Again, more frequent drought provides
the producer with incentives to prepare a crop plan in which drought is no longer
the unexpected outcome. These adjustments (e.g. leaving more fields open) allow
the producer to respond to more severe water shortages at less cost (e.g. fallowing
open fields, rather than fall-prepared fields).
Table 6.17. Average percent change in total discounted profit, by years of
drought experienced, for the base case and climate change cases 1 through
3.
Average % change in π
Years of Base Case Case Case
Drought Case*
1
2
3
1
-2
-1
-6
-6
*
2
-7
-3
-13
-11
3
-11
-7
-20
-18
4
-16
-11
-29
-26
5
-23
-16
-40
-35
6
-30
-23
-52
-45
Base case defined as Pr(D)=0.4, Dry = 24 ac-in/ac. Case 1 defined as
Pr(D)=0.5, Dry = 24 ac-in/ac. Case 2 defined as Pr(D)=0.4, Dry = 18 acin/ac. Case 3 defined as Pr(D)=0.5, Dry = 18 ac-in/ac.
190
The above results suggest that climate change has the potential to alter
agricultural producers’ need for government assistance in response to drought, but
in dramatically different ways, depending upon the characteristics of climate that
are affected. This result can help guide the evolution of farm support programs in
a changing climate. Specifically, producers will not be affected significantly by an
increase in drought frequency alone. In contrast, producers will be affected
significantly by an increase in drought severity or both severity and frequency.
Even after producers adjust cropping plans in response to these changes, they will
be much worse off during drought events than they are currently.
191
7
Summary of Results and Policy Implications
Chapter 7 summarizes the dissertation’s motivation, objective, method, and
major results. Chapter 7 also draws the large set of results together by discussing
their broader implications for the administration of drought-related farm programs,
in contrast to the farm-level implications presented in chapters 5 and 6. Readers
should note, again, that the term “optimal,” as used in this dissertation, refers
simply to activities that are included in the mathematical programming model’s
solution; it does not indicate that the activities are Pareto optimal or socially
efficient.
A transition from drought as ‘disaster’ to drought as ‘managed risk’ is
underway. The impact of recent severe droughts throughout the United States, the
potential for climate change to intensify the frequency and severity of drought, and
discussion about the future of government assistance in agriculture have all
increased the need to make this transition a reality. However, guidance for
agricultural producers about how to optimally manage for the risk of drought
remains insufficient.
Managing for the risk of drought involves two planning components,
drought preparedness and drought response. Optimal drought preparedness and
response is a challenging decision problem because few producers know a priori
whether drought will occur in the near future, when or how frequently it will occur
in the more distant future, how severe drought will be, or for how long any one
drought will persist. Producers whose farm systems are characterized by intra- and
inter-year dynamics face additional complexity in their decision environment.
They must consider not only how the outcomes of their decisions will vary across
states of nature, but also how their decisions today will affect opportunities and
outcomes in future periods. The future consequences of their current decisions
will also depend on the states of nature revealed through time. Given the
192
considerable complexity of this decision environment, it is not clear what optimal
drought preparedness and response, in practice, should look like.
A mathematical programming model that captures the stochastic and
dynamic nature of a representative irrigated mixed crop farm in eastern Oregon is
developed and used to explore the characteristics of optimal drought preparedness
and response. Insights are gained about the role of alternative preparedness and
response tools, the profit impact of droughts that vary in length, the tradeoff
between maximizing the use of scarce water resources and minimizing the effects
of discounting and interest costs, the role of crop history, the importance of interyear dynamics, the potential effects of climate change, the effectiveness of the
multi-peril crop insurance program’s prevented planting provision for reducing
drought’s impact on producer profit, and the influence of price uncertainty on the
management of water supply uncertainty. These insights, which are presented in
detail in chapters 5 and 6, are summarized below in the context of a broader
discussion of their potential implications for drought-related farm programs.
The magnitude of profit loss attributable to drought under optimal
preparedness and response increases as the number of years of drought increase. It
is difficult to generalize the impact of drought beyond this, however, because
profit loss exhibits large variation depending upon the year in which drought
occurs, or more specifically, the crops planted at the time the drought occurs. This
presents a challenge to the administration of drought-related farm programs.
Disaster declaration, for example, is based on the severity of loss at the county
level; that is, some threshold of loss must be reached at the county-level to qualify
for disaster assistance. This study illustrates that drought could impact identical
farms very differently if they are in a different stage of the crop plan when the
drought occurs. A farm community’s collective losses might therefore be
insufficient to receive disaster assistance, particularly if producers do not
synchronize their crop plans. Note, however, that the community’s economy is
193
likely sheltered from risk when producers do not synchronize their crop plans.
There are distributional implications of such a threshold-based policy; severely
impacted producers are likely to be left without assistance in all but the most
severe droughts. These severely impacted producers are not necessarily poor
managers, as is sometimes assumed. They could, in contrast, be optimally
prepared for drought, and simply have experienced an unfortunately-timed drought
with respect to their crop plan.
Crop insurance companies also judge the validity of a claim based in part
on the occurrence of similar losses in the neighboring area. It could be difficult for
crop insurance adjusters to identify comparable losses with which to validate a
producer’s claim. A producer with valid drought losses could potentially be
denied an indemnity payment because their crop plan is not synchronized with
their neighbors’. Timely program delivery would not be possible if assistance
were based on individual producers’ circumstances. However, farm program
administrators should recognize the potential for drought to generate
heterogeneous impacts, even across a set of homogeneous farms.
The potential for drought to generate spillover effects from one year to
another due to inter-year dynamics, a phenomenon which producers allude to, is
corroborated by the model’s results. A farm system with inter-year dynamics can
therefore continue to experience the effects of drought after the drought itself
subsides. Additionally, the effects of drought in one year can intensify the profit
impact of drought in subsequent years. Although producers likely prefer prompt
assistance in the event of drought, program administrators should keep in mind
that the total impact of a particular year of drought might not be felt for several
years, and that the impact of a multi-year drought can be more or less than the sum
of its parts. The marginal profit impact of a year of drought is shown for one
scenario, for example, to be 150% larger when preceded by a year of drought.
194
Producers face a complex set of tradeoffs when designing a drought
preparedness and response plan. One important tradeoff is between maximizing
the use of scarce water resources in each year of the planning horizon, and
minimizing the negative profit impacts of discount and interest rates. A producer
with a positive discount rate is generally expected to plant valuable crops first,
which implies, for this farm system, specialization in onions in year 1. However,
this strategy is not costless in a farm system with water supply uncertainty and
agronomic constraints that generate inter-year dynamics. It reduces the total
number of fields that can be successfully planted and the proportion of the total
water allotment used over the planning horizon. It is therefore not clear intuitively
whether the benefits of specialization outweigh the cost through time.
The model’s results indicate that specialization in onions in year 1 is
optimal, even in the presence of uncertainty and inter-year dynamics. This finding
differs from the commonly-held belief that diversification is an effective drought
management tool. It also contradicts the observed behavior of producers in the
study area, who tend to spread the production of high value crops, such as onions,
throughout the planning horizon, rather than concentrating it within one or two
years. A sensitivity analysis indicates, however, that this behavior is more likely
attributable to aversion to price-uncertainty for high-value crops, not water supply
uncertainty. For some producers, this behavior is also attributable to agronomic
constraints generated by past crop history.
Diversification is only optimal in the risk-neutral model when discount and
interest rates are set to zero, or drought is defined as very severe (a water allotment
of 12 acre-inches per acre, rather than 40). The hypothetical producer should,
more specifically, plant as many fields to the highest valued crop in the first year
as can be supported in the event of a drought. In the more probable case of less
severe drought (24 acre-inches per acre), up to seven fields can be supported
without risk of abandonment. Delaying the production of high value crops to reap
195
the benefits of more complete use of scarce water resources (i.e. more net revenue
from crop production) is not as profitable as concentrating the production early in
the planning horizon, due to the increased cost of discounting and interest on
borrowed funds. It would be difficult for a producer to weigh the relative
magnitude of these tradeoffs using intuition alone.
The potential for improved water supply forecasts to reduce the impacts of
water supply uncertainty receives much attention in the economics literature.
Losses attributable to drought can be separated into a portion caused by the actual
water shortage and a portion caused by uncertainty about the water supply. The
primary effect of water supply uncertainty in the farm system modeled here is that
more fall-prepared fields are abandoned than would be under certainty. This result
explains why the hypothetical producer is found to enroll in the multi-peril crop
insurance program’s prevented planting provision. The prevented planting
provision covers a portion of the losses incurred when an anticipated water
shortage makes it unreasonable to follow through with a planned crop. The
provision is shown to be an effective means for producers to prepare for and
mitigate the profit impacts of drought, even when premiums are not subsidized.
The social welfare implications of the provision are not examined, however.
The predicted effects of climate change for snowmelt-dependent farm
systems include more frequent and severe drought. Producers’ ability to formulate
and implement optimal drought preparedness and response plans will therefore
become increasingly important to the success of agricultural communities. The
general public relies on these communities to provide many public goods,
including open spaces and wildlife habitat. The public may also incur large
expenditures if agricultural producers suffer losses due to natural disasters, such as
drought, and receive assistance from the government. The ability of agricultural
producers to adapt optimally to a changing climate therefore has a variety of social
consequences. The impact of increased drought frequency and severity on optimal
196
drought preparedness and response, and on profit loss associated with drought is
analyzed. An increase in drought frequency has little impact on the drought
preparedness plan or on profit loss attributable to drought. However, an increase
in drought severity (or both severity and frequency) changes the relative
importance of alternative drought preparedness tools, and substantially increases
profit loss attributable to drought. The impact of climate change on the economic
sustainability of agricultural communities thus depends critically on which features
of the climate change. Future research on the impacts of climate change or the
value of improved climate change information should investigate alternative forms
of climate change to identify the range and uniqueness of optimal adjustments and
impacts.
The last body of literature to which this dissertation contributes is
mathematical modeling of stochastic and dynamic farm systems. The base case
model illustrates the use of multi-stage discrete sequential stochastic programming
(DSSP) to capture the dynamic and stochastic features of a farm system. Few
studies have taken advantage of multi-stage DSSP’s structure to represent both
intra- and inter-year dynamics. Because the model captures both intra- and interyear dynamics, it provides 1) a more thorough understanding of the complex
tradeoffs that producers face when preparing and responding to drought, 2) a more
complete picture of the impacts of drought through time, and 3) important insights
about the challenges that administrators of drought assistance programs face.
Lastly, it elucidates the applied aspects of optimal drought preparedness, a notion
that has received increased attention in the policy arena, but whose practical form
has been only vaguely discussed.
Beyond using DSSP to capture both intra- and inter-year dynamics, this
dissertation also contributes to the literature by solving both a binary and
continuous variables version of the model and comparing their solutions.
Continuous variables are commonly used, rather than binary variables, because
197
linear programming models are more easily solved than integer programming
models, particularly when the model is stochastic. Many farm decisions, in
contrast, are innately binary; for example, crops are often chosen for individual
fields, rather than individual acres, or portions of an acre. A binary model
represents the producer’s decision problem more accurately than a continuous
model, but it is also more difficult to solve. The ability of a continuous model to
approximate the binary model’s solution is thus examined.
The continuous model suggests similar cropping activities and timing.
However, because the continuous model affords more flexibility in activities, it
does not identify the same set of drought preparedness and response tools as the
binary model. More importantly, drought preparedness tools that are observed on
the ground are identified by the binary model, but not by the continuous model.
Had the continuous model been used exclusively, one might have concluded that
the activities excluded from the solution are sub-optimal, when they are in fact
optimal in a more realistic model. In conclusion, factors such as field size or other
discontinuities in the farm system need to be considered when choosing between a
continuous versus binary (or integer) model.
The results of this dissertation shed light on several important aspects of
drought management at the farm-level. However, there are many opportunities to
improve and expand this analysis. A need exists to more realistically capture
constraints on the producer’s ability to change irrigation technologies from yearto-year, or to accommodate alternative scales of production for individual crops
(i.e. machinery and labor constraints). The producer’s crop mix and irrigation
technology is more constrained than is assumed here; hence, expected profit and
profit in dry years are likely less than those found here. A more accurate
mathematical representation of yield response to deficit irrigation is also needed,
such that strategic rather than season-long deficits can be modeled. Deficit
irrigation plays a larger role in the response to drought than is found here.
198
Strategic deficit irrigation may enable producers to adjust their crop mix such that
expected profit and profit in dry years are increased.
More work is also needed to better represent the continuous and updating
nature of decision-making through time. Terminal values are included in the
existing model to reflect that the farm continues to operate after the current
planning horizon ends. However, explicit modeling of decisions in subsequent
planning horizons would capture the long-term dynamics directly. This would
also enable a more thorough analysis of the impact of crop history on the optimal
drought preparedness strategy and the transition to that strategy. More advanced
programming and a more powerful solution algorithm would be needed, however,
to accommodate such additional stages in the decision problem. Additional farm
programs should also be included in subsequent modeling efforts, such as loan
deficiency payments, low-interest rate loans, and the conservation reserve
program, to place drought preparedness and response tools in a more realistic
decision context. Lastly, the modeling framework used here should be applied to
other farm systems that exhibit inter-year dynamics to determine whether optimal
drought preparedness and response, and the impacts of drought in these systems
are affected by inter-year dynamics in ways similar to the farm system modeled in
this dissertation.
199
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Appendices
215
Appendix A. Base case parameters and profit
Table A.1. Per-unit prices and maximum yield assumed in the base case.
Crop (yield unit)
Onion (cwt)
Russet Potato (cwt)
Sugar Beet (T)
Winter Wheat (bu)
Alfalfa Establishment (T)
Alfalfa Established (T)
Grain Corn (bu)
Silage Corn (T)
Barley (bu)
1
Price
per Unit
($2004)1
6.00
3.30
39.00
3.20
79.00
79.00
2.70
19.00
2.15
Max Yield
(irrigation
system)2
650 (drip)
415 (furrow)
31 (furrow)
130 (furrow)
6 (furrow)
6 (furrow)
170 (furrow)
28 (furrow)
100 (furrow)
Ten-year historical average price (1995-2004) as reported by the Malheur County
Extension Service (2004a). 2 Maximum yield based on Malheur County Extension
Service (2004a) and conversations with producers in the study area.
216
Table A.2. Cost ($2004/acre) of crop production under alternative irrigation
technologies.
Fall Cost
Spring Cost
Opp Cost
of $1
Econ
Cost2
600
600
600
2400
2480
2750
210
216
235
3210
3296
3585
100
100
100
1350
1450
1625
102
109
121
1552
1659
1846
150
150
150
150
820
920
940
940
68
75
76
76
1038
1145
1166
1166
Furrow
Reuse furrow
Wheel-line sprinkler
Center pivot sprinkler
160
160
160
160
105
145
190
190
19
21
25
25
284
326
375
375
Alfalfa Establishment
Furrow
Reuse furrow
Wheel-line sprinkler
Center pivot sprinkler
185
185
185
185
295
330
380
380
34
36
40
40
514
551
605
605
Alfalfa Established
Furrow
Reuse furrow
Wheel-line sprinkler
Center pivot sprinkler
0
0
0
0
295
330
380
380
21
23
27
27
316
353
407
407
----
425
465
510
30
33
36
455
498
546
Crop/Irrigation
Onion
Furrow
Reuse furrow
Drip
Russet Potato
Furrow
Reuse furrow
Solid set sprinkler
Sugar Beet
Furrow
Reuse furrow
Wheel-line sprinkler
Center pivot sprinkler
Winter Wheat
Grain Corn
Furrow
Reuse furrow
Center pivot sprinkler
217
Table A.2 (Cont.)
Crop/Irrigation
Fall Cost
Spring Cost
Opp Cost1
Econ Cost2
----
600
640
685
42
45
48
642
685
733
------
245
285
330
330
0
17
20
23
23
0
262
305
353
353
0
Silage Corn
Furrow
Reuse furrow
Center pivot sprinkler
Barley
Furrow
Reuse furrow
Wheel-line sprinkler
Center pivot sprinkler
Fallow
1
Equals 1.07 times the sum of Fall and Spring Costs (i.e. 7% is the rate earned on
own funds if saved rather than spent, or the interest rate on borrowed funds).
2
Equals the sum of “Fall Cost,” “Spring Cost,” and “Opp Cost.”
Enterprise budgets compiled for Malheur County crops were the primary source of
cost data. Specific sources include the following: (Boswell et al. 1995; Malheur
County Extension Service 2004b; Malheur County Extension Service 2004c;
Malheur County Extension Service 2004d; Malheur County Extension Service
2000; Malheur County Extension Service 2002; Malheur County Extension
Service 2004e; Malheur County Extension Service 2003; Turner and Bohle 1995).
All cost data were adjusted for inflation to 2004$ using the Prices Paid Index
(Crop Sector) for Commodities & Services, Interest, Taxes & Wage Rates. The
following sources were used to adapt enterprise budgets for alternative irrigation
technologies: (Hinman et al. 1997; Klauzer 2005; Patterson, King, and Smathers
1996a; Patterson, King, and Smathers 1996b; Smathers, King, and Patterson
1995).
218
Table A.3. Crop water requirements.
Crop
Onion
Russet Potato
Sugar Beet
Winter Wheat
Alfalfa Establishment
Alfalfa Established
Grain Corn
Silage Corn
Barley
Fallow
*
Crop Water Requirement
(inches)*
29.0
27.2
34.1
24.1
41.8
41.8
27.5
27.5
26.1
0
Crop water requirement is assumed equal to the 13-year average (1992-2004)
seasonal evapotranspiration for the respective crop at the Ontario, Oregon Agrimet
station (Bureau of Reclamation 2006). Average ET reflects the crop water
requirement that must be met to produce the average yield observed in the study
area.
219
Table A.4. Technical efficiency (%) of alternative irrigation technologies.
Irrigation
Technology
Furrow
Reuse furrow
Solid set sprinkler
Wheel-line sprinkler
Center pivot sprinkler
Subsurface drip
*
Technical
efficiency*
50
80
65
65
75
90
Technical efficiency is defined as the proportion of water delivered to the crop
that reaches the crop root zone (i.e. the proportion of delivered water that does not
runoff, evaporate or percolate out of the root zone). Sources: (Hoffman and
Willett 1998; Neibling 1997; Oregon State University Water Resources
Engineering Team 1992, p179)
220
Table A.5. Assumed yield response factor (ky) for alternative crops.
Crop
Onion
Russet Potato
Sugar Beet
Winter Wheat
Alfalfa Establishment
Alfalfa Established
Grain Corn
Silage Corn
Barley
*
Yield Response
Factor (ky)*
1.10
1.10
0.80
1.00
0.90
0.90
1.25
1.25
1.00
Yield response factors indicate sensitivity of yield to a water deficit of equal
proportion throughout the growing season (in contrast to a water deficit during a
particular growth stage). ky > 1 indicate relatively drought intolerant crops; ky ≤ 1
indicate relatively drought tolerant crops. Source: Doorenbos and Kassam (1979,
table 24).
221
Table A.6. Crop yield (units/acre), total revenue ($/acre), total cost ($/acre), and
net revenue ($/acre) for alternative combinations of crop, irrigation technology,
and deficit irrigation level.
Crop/Irrigation
Deficit1
Yield
Tot Rev
Tot Cost2
Net Rev
Onion (cwt)
Furrow
Reuse Furrow
Drip
Russet Potato (cwt)
Furrow
Reuse Furrow
Solid Set Sprinkler
Sugar Beet (T)
Furrow
D1
D2
D3
D4
D5
D6
D1
D2
D3
D4
D5
D6
D1
D2
D3
D4
D5
D6
550
498
446
394
341
289
550
498
446
394
341
289
650
588
527
465
403
342
3300
2987
2674
2361
2048
1735
3300
2987
2674
2361
2048
1735
3900
3530
3160
2791
2421
2051
3210
3210
3210
3210
3210
3210
3296
3296
3296
3296
3296
3296
3585
3585
3585
3585
3585
3585
90
-223
-536
-849
-1162
-1475
4
-309
-621
-934
-1247
-1560
316
-54
-424
-794
-1164
-1534
D1
D1
D1
415
415
450
1370
1370
1485
1552
1659
1846
-182
-289
-361
D1
31
1209
1038
171
D2
29
1124
1038
86
D3
27
1038
1038
0
D4
24
953
1038
-85
D5
22
867
1038
-171
D6
20
782
1038
-256
1
Represents the proportion of irrigation water requirement provided (D1=100%,
D2=90%,…, D6=50%,D7=0%). 2Includes a 7% opportunity cost of money;
excludes opportunity cost of land and management.
222
Table A.6 (Cont.)
Crop/Irrigation
Sugar Beet (T)
Reuse Furrow
Wheel Line
Center Pivot
Winter Wheat (bu)
Furrow
Reuse Furrow
Deficit
Yield
Tot Rev
Tot Cost
Net Rev
D1
D2
D3
D4
D5
D6
D1
D2
D3
D4
D5
D6
D1
D2
D3
D4
D5
D6
31
29
27
24
22
20
31
29
27
24
22
20
26
24
22
20
19
17
1209
1124
1038
953
867
782
1209
1124
1038
953
867
782
1014
942
871
799
728
656
1145
1145
1145
1145
1145
1145
1166
1166
1166
1166
1166
1166
1166
1166
1166
1166
1166
1166
64
-21
-107
-192
-278
-363
43
-43
-128
-214
-299
-384
-152
-224
-295
-367
-439
-510
D1
D2
D3
D4
D5
D6
D1
D2
D3
D4
D5
D6
130
119
108
97
87
76
130
119
108
97
87
76
416
381
347
312
277
243
416
381
347
312
277
243
284
284
284
284
284
284
326
326
326
326
326
326
132
98
63
28
-6
-41
90
55
20
-14
-49
-84
223
Table A.6 (Cont.)
Crop/Irrigation
Winter Wheat (bu)
Wheel Line
Center Pivot
Alfalfa-1st yr (T)
Furrow
Reuse Furrow
Wheel Line
Deficit
Yield
Tot Rev
Tot Cost
Net Rev
D1
D2
D3
D4
D5
D6
D1
D2
D3
D4
D5
D6
130
119
108
97
87
76
110
101
92
82
73
64
416
381
347
312
277
243
352
323
293
264
235
205
375
375
375
375
375
375
375
375
375
375
375
375
42
7
-28
-63
-97
-132
-23
-52
-81
-111
-140
-169
D1
D2
D3
D4
D5
D6
D1
D2
D3
D4
D5
D6
D1
D2
D3
D4
D5
D6
6
6
5
5
4
4
6
6
5
5
4
4
7
6
6
5
5
4
474
435
397
359
320
281
474
435
397
359
320
281
553
508
463
418
373
328
514
514
514
514
514
514
551
551
551
551
551
551
605
605
605
605
605
605
-40
-78
-117
-155
-194
-232
-77
-116
-154
-192
-231
-270
-52
-97
-142
-187
-232
-277
224
Table A.6 (Cont.)
Crop/Irrigation
Alfalfa-1st yr (T)
Center Pivot
Alfalfa-yrs 2-4 (T)
Furrow
Reuse Furrow
Wheel Line
Center Pivot
Deficit
Yield
Tot Rev
Tot Cost
Net Rev
D1
D2
D3
D4
D5
D6
7
6
5
5
4
4
514
472
430
388
346
304
605
605
605
605
605
605
-91
-133
-175
-217
-259
-300
D1
D2
D3
D4
D5
D6
D1
D2
D3
D4
D5
D6
D1
D2
D3
D4
D5
D6
D1
D2
D3
D4
D5
D6
6
6
5
5
4
4
6
6
5
5
4
4
7
6
6
5
5
4
7
6
5
5
4
4
474
435
397
359
320
281
474
435
397
359
320
281
553
508
463
418
373
328
514
472
430
388
346
304
316
316
316
316
316
316
353
353
353
353
353
353
407
407
407
407
407
407
407
407
407
407
407
407
158
120
81
43
4
-34
121
82
43
6
-33
-72
146
101
56
11
-34
-79
107
65
23
-19
-61
-102
225
Table A.6 (Cont.)
Crop/Irrigation
Grain Corn (bu)
Furrow
Reuse Furrow
Center Pivot
Silage Corn (T)
Furrow
Reuse Furrow
Center Pivot
Deficit
Yield
Tot Rev
Tot Cost
Net Rev
D1
D2
D3
D1
D2
D3
D1
D2
D3
170
152
134
170
152
134
180
161
142
459
410
361
459
410
361
486
434
382
455
455
455
498
498
498
546
546
546
4
-45
-94
-39
-88
-137
-60
-112
-164
D1
D2
D3
D1
D2
D3
D1
D2
D3
28
25
22
28
25
22
30
27
24
532
475
418
532
475
418
570
509
448
642
642
642
685
685
685
733
733
733
-110
-167
-224
-153
-210
-266
-163
-224
-285
D1
D2
D3
D4
D5
D6
D1
D2
D3
D4
D5
D6
100
92
83
75
66
58
100
92
83
75
66
58
215
197
179
160
142
124
215
197
179
160
142
124
262
262
262
262
262
262
305
305
305
305
305
305
-47
-65
-84
-102
-120
-138
-90
-108
-126
-145
-163
-181
Barley (bu)
Furrow
Reuse Furrow
226
Table A.6 (Cont.)
Crop/Irrigation
Barley (bu)
Wheel Line
Center Pivot
Deficit
Yield
Tot Rev
Tot Cost
Net Rev
D1
D2
D3
D4
D5
D6
D1
D2
D3
D4
D5
D6
100
92
83
75
66
58
90
82
75
67
60
52
215
197
179
160
142
124
194
177
161
144
128
112
353
353
353
353
353
353
353
353
353
353
353
353
-138
-156
-174
-193
-211
-229
-160
-176
-192
-209
-225
-242
D7
0
0
0
0
Fallow
Furrow
227
Table A.7. Discounted and undiscounted profit (i.e. returns to land and
management) ($2004) for each water supply scenario in the binary base case
solution. Scenarios are grouped by the number of years of drought experienced,
and then sorted within each group by discounted profit (in ascending order).
Yr1
Yr2
Yr3
Yr4
Yr5
Yr6
Discounted π
Undiscounted π
6 Years of Drought
DRY DRY DRY
DRY
DRY
DRY
408,273
594,431
5 Years of Drought
FULL DRY DRY
DRY DRY DRY
DRY DRY DRY
DRY DRY DRY
DRY FULL DRY
DRY DRY FULL
DRY
DRY
DRY
FULL
DRY
DRY
DRY
FULL
DRY
DRY
DRY
DRY
DRY
DRY
FULL
DRY
DRY
DRY
437,473
443,160
445,155
452,224
462,412
468,926
630,416
642,622
646,684
654,089
664,657
674,699
4 Years of Drought
FULL DRY DRY
DRY DRY DRY
DRY DRY DRY
DRY FULL DRY
FULL DRY DRY
DRY DRY DRY
DRY DRY FULL
FULL DRY DRY
DRY DRY FULL
DRY FULL DRY
FULL DRY FULL
FULL FULL DRY
DRY FULL DRY
DRY DRY FULL
DRY FULL FULL
DRY
FULL
DRY
DRY
DRY
FULL
DRY
FULL
DRY
DRY
DRY
DRY
FULL
FULL
DRY
DRY
DRY
FULL
DRY
FULL
FULL
DRY
DRY
FULL
FULL
DRY
DRY
DRY
DRY
DRY
FULL
FULL
FULL
FULL
DRY
DRY
FULL
DRY
DRY
DRY
DRY
DRY
DRY
DRY
DRY
462,219
466,120
467,906
476,308
477,091
477,820
481,535
482,531
484,284
486,918
495,857
501,189
505,119
523,732
525,277
665,474
673,775
677,680
684,343
684,755
689,196
692,563
691,477
695,764
698,269
706,903
712,094
722,246
748,328
746,712
228
Table A.7 (Cont.)
Yr1
Yr2
Yr3
Yr4
Yr5
Yr6
3 Years of Drought
FULL DRY DRY
DRY DRY DRY
DRY DRY FULL
FULL DRY FULL
DRY FULL DRY
FULL DRY DRY
DRY FULL DRY
FULL DRY DRY
DRY FULL DRY
FULL DRY FULL
FULL FULL DRY
DRY DRY FULL
FULL FULL DRY
DRY DRY FULL
FULL FULL DRY
FULL FULL FULL
DRY FULL FULL
DRY FULL FULL
DRY FULL FULL
FULL DRY FULL
FULL
FULL
DRY
DRY
DRY
DRY
FULL
FULL
FULL
DRY
FULL
FULL
DRY
FULL
DRY
DRY
DRY
DRY
FULL
FULL
DRY
FULL
FULL
FULL
FULL
FULL
DRY
FULL
FULL
DRY
DRY
DRY
DRY
FULL
FULL
DRY
FULL
DRY
DRY
DRY
FULL
FULL
FULL
DRY
FULL
FULL
FULL
DRY
DRY
FULL
DRY
FULL
FULL
DRY
DRY
DRY
DRY
FULL
DRY
DRY
Discounted π
Undiscounted π
486,863
491,716
496,893
500,644
500,814
501,836
506,878
509,880
519,054
520,603
523,933
524,205
525,935
526,112
528,539
531,460
536,474
537,886
546,214
554,160
697,613
708,882
713,628
713,469
717,955
719,813
724,738
728,989
741,360
741,961
742,845
748,998
747,152
751,548
749,606
751,204
762,236
764,575
774,853
785,537
229
Table A.7 (Cont.)
Yr1
Yr2
Yr3
Yr4
Yr5
Yr6
Discounted π Undiscounted π
2 Years of Drought
FULL DRY DRY
DRY FULL DRY
FULL DRY FULL
DRY DRY FULL
FULL FULL DRY
DRY FULL FULL
FULL FULL FULL
FULL FULL DRY
DRY FULL FULL
FULL FULL DRY
DRY FULL FULL
FULL DRY FULL
FULL FULL FULL
FULL DRY FULL
FULL FULL FULL
FULL
FULL
DRY
FULL
FULL
DRY
DRY
FULL
FULL
DRY
FULL
FULL
DRY
FULL
FULL
FULL
FULL
FULL
FULL
DRY
FULL
FULL
FULL
DRY
FULL
FULL
DRY
DRY
FULL
DRY
FULL
FULL
FULL
FULL
FULL
FULL
DRY
DRY
FULL
FULL
DRY
FULL
FULL
DRY
DRY
514,212
520,813
525,390
529,157
536,542
540,806
544,149
551,436
552,818
553,284
554,124
554,633
556,206
557,526
576,205
735,126
743,852
748,527
755,862
760,709
768,373
768,607
780,575
784,209
784,664
785,907
786,207
786,262
790,154
811,609
1 Years of Drought
DRY FULL FULL
FULL DRY FULL
FULL FULL DRY
FULL FULL FULL
FULL FULL FULL
FULL FULL FULL
FULL
FULL
FULL
DRY
FULL
FULL
FULL
FULL
FULL
FULL
FULL
DRY
FULL
FULL
FULL
FULL
DRY
FULL
554,597
557,998
564,045
568,894
579,658
590,100
786,577
790,823
798,439
803,665
816,549
831,295
0 Years of Drought
FULL FULL FULL FULL FULL FULL
582,703
820,863
230
Table A.8. Irrigation water requirement and net revenue ($) per acre-inch of water
requirement for alternative crop-irrigation-deficit combinations.
Crop,
Irrig Tech &
Deficit Level
Onion (cwt)
Furrow
D1
D2
D3
D4
D5
D6
Reuse Furrow
D1
D2
D3
D4
D5
D6
Drip
D1
D2
D3
D4
D5
D6
Irrig
Water
Req1
Proportion
Irrig
Irrig
2
3
Provided Efficiency Delivery4
Net Rev/
acre-inch
25
25
25
25
25
25
1
0.9
0.8
0.7
0.6
0.5
0.5
0.5
0.5
0.5
0.5
0.5
50.00
45.00
40.00
35.00
30.00
25.00
1.80
-4.95
-13.40
-24.25
-38.72
-58.99
25
25
25
25
25
25
1
0.9
0.8
0.7
0.6
0.5
0.8
0.8
0.8
0.8
0.8
0.8
31.25
28.13
25.00
21.88
18.75
15.63
0.14
-10.97
-24.86
-42.72
-66.52
-99.86
25
25
25
25
25
25
1
0.9
0.8
0.7
0.6
0.5
0.9
0.9
0.9
0.9
0.9
0.9
27.78
25.00
22.22
19.44
16.67
13.89
11.36
-2.17
-19.09
-40.83
-69.83
-110.42
Rus. Potato (cwt)
Furrow
D1
23.2
1
0.5
46.40
-3.92
Reuse Furrow
D1
23.2
1
0.8
29.00
-9.97
1
2
Crop water requirement less 4” of effective precipitation. Proportion of
requirement supplied. 3Proportion of delivered water that reaches crop root zone.
4
Actual water delivery required to meet water requirements, given the proportion
provided (i.e. deficit level) and irrigation efficiency.
231
Table A.8 (Cont.)
Crop,
Irrig Tech &
Deficit Level
Irrig
Water
Req
Proportion
Provided
Irrig
Efficiency
Rus. Potato (cwt)
Solid Set
D1
23.2
1
0.65
35.69
-10.11
30.1
30.1
30.1
30.1
30.1
30.1
1
0.9
0.8
0.7
0.6
0.5
0.5
0.5
0.5
0.5
0.5
0.5
60.20
54.18
48.16
42.14
36.12
30.10
2.84
1.58
0.01
-2.02
-4.72
-8.50
30.1
30.1
30.1
30.1
30.1
30.1
1
0.9
0.8
0.7
0.6
0.5
0.8
0.8
0.8
0.8
0.8
0.8
37.63
33.86
30.10
26.34
22.58
18.81
1.70
-0.63
-3.55
-7.29
-12.29
-19.29
30.1
30.1
30.1
30.1
30.1
30.1
1
0.9
0.8
0.7
0.6
0.5
0.65
0.65
0.65
0.65
0.65
0.65
46.31
41.68
37.05
32.42
27.78
23.15
0.92
-1.02
-3.46
-6.59
-10.76
-16.60
30.1
30.1
30.1
30.1
30.1
30.1
1
0.9
0.8
0.7
0.6
0.5
0.75
0.75
0.75
0.75
0.75
0.75
40.13
36.12
32.11
28.09
24.08
20.07
-3.79
-6.20
-9.20
-13.07
-18.21
-25.43
Irrig
Delivery
Net Rev/
acre-inch
Sugar Beet (T)
Furrow
D1
D2
D3
D4
D5
D6
Reuse Furrow
D1
D2
D3
D4
D5
D6
Wheel Line
D1
D2
D3
D4
D5
D6
Center Pivot
D1
D2
D3
D4
D5
D6
232
Table A.8 (Cont.)
Crop,
Irrig Tech &
Deficit Level
Winter Wheat (bu)
Furrow
D1
D2
D3
D4
D5
D6
Reuse Furrow
D1
D2
D3
D4
D5
D6
Wheel Line
D1
D2
D3
D4
D5
D6
Center Pivot
D1
D2
D3
D4
D5
D6
Irrig
Water
Req
Proportion
Provided
Irrig
Efficiency
20.1
20.1
20.1
20.1
20.1
20.1
1
0.9
0.8
0.7
0.6
0.5
0.5
0.5
0.5
0.5
0.5
0.5
40.20
36.18
32.16
28.14
24.12
20.10
3.29
2.70
1.96
1.01
-0.26
-2.04
20.1
20.1
20.1
20.1
20.1
20.1
1
0.9
0.8
0.7
0.6
0.5
0.8
0.8
0.8
0.8
0.8
0.8
25.13
22.61
20.10
17.59
15.08
12.56
3.57
2.43
1.01
-0.82
-3.26
-6.67
20.1
20.1
20.1
20.1
20.1
20.1
1
0.9
0.8
0.7
0.6
0.5
0.65
0.65
0.65
0.65
0.65
0.65
30.92
27.83
24.74
21.65
18.55
15.46
1.34
0.24
-1.13
-2.89
-5.24
-8.54
20.1
20.1
20.1
20.1
20.1
20.1
1
0.9
0.8
0.7
0.6
0.5
0.75
0.75
0.75
0.75
0.75
0.75
26.80
24.12
21.44
18.76
16.08
13.40
-0.84
-2.15
-3.79
-5.89
-8.70
-12.63
Irrig
Delivery
Net Rev/
acre-inch
233
Table A.8 (Cont.)
Crop,
Irrig Tech &
Deficit Level
Alfalfa-1st yr (T)
Furrow
D1
D2
D3
D4
D5
D6
Reuse Furrow
D1
D2
D3
D4
D5
D6
Wheel Line
D1
D2
D3
D4
D5
D6
Center Pivot
D1
D2
D3
D4
D5
D6
Irrig
Water
Req
Proportion
Provided
Irrig
Efficiency
37.8
37.8
37.8
37.8
37.8
37.8
1
0.9
0.8
0.7
0.6
0.5
0.5
0.5
0.5
0.5
0.5
0.5
75.60
68.04
60.48
52.92
45.36
37.80
-0.52
-1.15
-1.93
-2.93
-4.27
-6.15
37.8
37.8
37.8
37.8
37.8
37.8
1
0.9
0.8
0.7
0.6
0.5
0.8
0.8
0.8
0.8
0.8
0.8
47.25
42.53
37.80
33.08
28.35
23.63
-1.63
-2.72
-4.09
-5.82
-8.15
-11.42
37.8
37.8
37.8
37.8
37.8
37.8
1
0.9
0.8
0.7
0.6
0.5
0.65
0.65
0.65
0.65
0.65
0.65
58.15
52.34
46.52
40.71
34.89
29.08
-0.89
-1.85
-3.04
-4.58
-6.64
-9.52
37.8
37.8
37.8
37.8
37.8
37.8
1
0.9
0.8
0.7
0.6
0.5
0.75
0.75
0.75
0.75
0.75
0.75
50.40
45.36
40.32
35.28
30.24
25.20
-1.81
-2.93
-4.34
-6.14
-8.55
-11.92
Irrig
Delivery
Net Rev/
acre-inch
234
Table A.8 (Cont.)
Crop,
Irrig Tech &
Deficit Level
Alfalfa-yrs 2-4 (T)
Furrow
D1
D2
D3
D4
D5
D6
Reuse Furrow
D1
D2
D3
D4
D5
D6
Wheel Line
D1
D2
D3
D4
D5
D6
Center Pivot
D1
D2
D3
D4
D5
D6
Irrig
Water
Req
Proportion
Provided
Irrig
Efficiency
37.8
37.8
37.8
37.8
37.8
37.8
1
0.9
0.8
0.7
0.6
0.5
0.5
0.5
0.5
0.5
0.5
0.5
75.60
68.04
60.48
52.92
45.36
37.80
2.09
1.76
1.34
0.81
0.09
-0.91
37.8
37.8
37.8
37.8
37.8
37.8
1
0.9
0.8
0.7
0.6
0.5
0.8
0.8
0.8
0.8
0.8
0.8
47.25
42.53
37.80
33.08
28.35
23.63
2.56
1.93
1.15
0.17
-1.17
-3.04
37.8
37.8
37.8
37.8
37.8
37.8
1
0.9
0.8
0.7
0.6
0.5
0.65
0.65
0.65
0.65
0.65
0.65
58.15
52.34
46.52
40.71
34.89
29.08
2.52
1.94
1.21
0.28
-0.97
-2.71
37.8
37.8
37.8
37.8
37.8
37.8
1
0.9
0.8
0.7
0.6
0.5
0.75
0.75
0.75
0.75
0.75
0.75
50.40
45.36
40.32
35.28
30.24
25.20
2.12
1.43
0.57
-0.53
-2.00
-4.07
Irrig
Delivery
Net Rev/
acre-inch
235
Table A.8 (Cont.)
Crop,
Irrig Tech &
Deficit Level
Grain Corn (bu)
Furrow
D1
D2
D3
Reuse Furrow
D1
D2
D3
Center Pivot
D1
D2
D3
Silage Corn (T)
Furrow
D1
D2
D3
Reuse Furrow
D1
D2
D3
Center Pivot
D1
D2
D3
Irrig
Water
Req
Proportion
Provided
Irrig
Efficiency
23.5
23.5
23.5
1
0.9
0.8
23.5
23.5
23.5
Irrig
Delivery
Net Rev/
acre-inch
0.5
0.5
0.5
47.00
42.30
37.60
0.09
-1.06
-2.50
1
0.9
0.8
0.8
0.8
0.8
29.38
26.44
23.50
-1.31
-3.31
-5.81
23.5
23.5
23.5
1
0.9
0.8
0.75
0.75
0.75
31.33
28.20
25.07
-1.91
-3.96
-6.52
23.5
23.5
23.5
1
0.9
0.8
0.5
0.5
0.5
47.00
42.30
37.60
-2.34
-3.94
-5.95
23.5
23.5
23.5
1
0.9
0.8
0.8
0.8
0.8
29.38
26.44
23.50
-5.20
-7.93
-11.34
23.5
23.5
23.5
1
0.9
0.8
0.75
0.75
0.75
31.33
28.20
25.07
-5.20
-7.93
-11.36
236
Table A.8 (Cont.)
Crop,
Irrig Tech &
Deficit Level
Irrig
Water
Req
Proportion
Provided
Irrig
Efficiency
22.1
22.1
22.1
22.1
22.1
22.1
1
0.9
0.8
0.7
0.6
0.5
22.1
22.1
22.1
22.1
22.1
22.1
Irrig
Delivery
Net Rev/
acre-inch
0.5
0.5
0.5
0.5
0.5
0.5
44.20
39.78
35.36
30.94
26.52
22.10
-1.07
-1.64
-2.36
-3.29
-4.52
-6.25
1
0.9
0.8
0.7
0.6
0.5
0.8
0.8
0.8
0.8
0.8
0.8
27.63
24.86
22.10
19.34
16.58
13.81
-3.26
-4.35
-5.72
-7.48
-9.82
-13.10
22.1
22.1
22.1
22.1
22.1
22.1
1
0.9
0.8
0.7
0.6
0.5
0.65
0.65
0.65
0.65
0.65
0.65
34.00
30.60
27.20
23.80
20.40
17.00
-4.06
-5.11
-6.42
-8.10
-10.34
-13.48
22.1
22.1
22.1
22.1
22.1
22.1
1
0.9
0.8
0.7
0.6
0.5
0.75
0.75
0.75
0.75
0.75
0.75
29.47
26.52
23.57
20.63
17.68
14.73
-5.42
-6.64
-8.16
-10.12
-12.73
-16.39
0
0
0.5
0.00
0.00
Barley (bu)
Furrow
D1
D2
D3
D4
D5
D6
Reuse Furrow
D1
D2
D3
D4
D5
D6
Wheel Line
D1
D2
D3
D4
D5
D6
Center Pivot
D1
D2
D3
D4
D5
D6
Fallow
Furrow
D7
237
Appendix B. Description of the Gaussian quadrature procedure used to assign
water quantities and probabilities (and onion prices and probabilities) to each state
of nature.
Miller and Rice (1983) show that approximating a continuous distribution
with a discrete number of categories that are defined as the midpoints of intervals
of interest underestimates the true distribution’s higher moments. They suggest an
alternative way to define the discrete categories, using Gaussian quadrature. This
approach is generally able to maintain the distributions more accurately than the
standard midpoint approach; however, the accuracy of the Gaussian approximation
varies by distribution. Preckel and DeVuyst (1992) also demonstrate that this
approach performs better than alternative methods and recommend its use in
discrete stochastic programming models.
Miller and Rice (1983) present two approaches. One approach is for cases
in which the random variable’s continuous distribution and moments are thought
to be known; the other approach is for cases in which the distribution is not
known. Each approach is described briefly and illustrated using historical water
allocation data from the study area.
B.1 Distribution Known:
Cases in which the random variable’s distribution is thought to be known
involve taking the Gaussian quadrature of the distribution for a chosen N. N is the
desired number of discrete categories with which to represent the continuous
distribution. The statistical software package R can perform Gaussian quadrature
calculations. Output from the Gaussian quadrature includes N pairs of values and
probabilities, which respectively define and assign probabilities to each category.
The random variable, water allotment, is naturally censored below by zero
and above by the maximum reservoir storage capacity. The beta distribution is the
only distribution, to the author’s knowledge, that allows censoring on both ends,
so it was considered as a possible distribution for the random variable. The fit of
238
the beta distribution needed to be tested first. Water allotment data for the study
area from 1981-2004 was used to create a relative frequency histogram. Note that
the beta distribution is typically expressed with bounds of [0,1]; data has to be
adjusted to this scale before applying the distribution. Parameters for the beta
distribution were first estimated from the data (Haan 2002, p141). The beta
distribution’s probability distribution function was then constructed and overlaid
on the empirical relative frequency histogram (figure B.1). The degree of fit was
visually judged. The beta distribution captures the major characteristics of the
histogram. Proceeding, for illustrative purposes, on the assumption that the data
comes from the beta distribution, a Gaussian quadrature of the parameterized beta
distribution is performed for N = 2 and 3. The suggested values and probabilities
for N = 2 are (16”, 40%), and (39”, 60%). The suggested values and probabilities
Frequency
for N = 3 are (10”, 16%), (27”, 49%), and (43”, 35%).
Frequency
f(x)beta
Bin
Figure B.1. Beta distribution’s probability distribution function
(parameterized with historical water allotment data from the study area),
overlaid on a relative frequency histogram of the data.
239
B.2 Distribution Unknown
Cases in which the random variable’s distribution is completely unknown
require an alternative approach. A cumulative distribution function (cdf) is first
estimated from the data. Table 3 in Miller and Rice (1983) is then used as follows:
choose the desired N; find the F(x) associated with N in table 3; identify the x in
the empirical cdf that correspond to the recommended F(x); assign to each x the
probability suggested in table 3 for each F(x). Proceeding on the assumption that
the water allotment’s distribution is unknown, a cdf was estimated from the
historical data (figure B.2). The suggested values and probabilities for N = 2 are
(24”, 50%) and (41”, 50%), and those for N = 3 are (12”, 25%), (36”, 50%), (42”,
50%).
1.1
1
0.9
0.8
F(x)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Water Supply (acre-inches per acre)
Figure B.2. Cumulative distribution function for the random variable
“water supply,” estimated with historical water allotment data from the
study area.
4.5
240
Prior to being aware of this approach, interviews with producers were used
to select economically meaningful values for the categories, “FULL” and “dry”.
Those values were identified as 24” and 40”. Historical data was used to assign
probabilities of 40% and 60%, respectively, to the categories. The Gaussian
quadrature approach conveniently suggested nearly the same values, and similar
probabilities. Prior analyses had been conducted at the 40 and 60% probabilities.
One model was resolved with 50% probabilities in place, and the results were not
significantly different. The cost of re-running all analyses with 50% probabilities
in place outweighed the benefits, so the original values and probabilities were
kept.
Miller and Rice’s approach does not perfectly preserve all moments of the
distribution. They report that for a particular beta distribution (parameter values
are not reported) the Gaussian approach (for N=2) overestimates variance by
10.5%, underestimates skew by 100%, and underestimates kurtosis by 48.3%.
This performance is better, however, than the standard midpoint approach, which
underestimates variance by 31.5%, underestimates skew by 100%, and
underestimates kurtosis by 80.1%. The degree of improvement varies, however,
by distribution.
B.3 Discrete Categories for the Onion Price
Proceeding on the assumption that onion price’s distribution is unknown, a
cdf was estimated from the historical data (figure B.3). The suggested values and
probabilities for N = 2 are ($2.80, 50%) and ($8.75, 50%), and those for N = 3 are
($2.50, 25%), ($5.50, 50%), ($12.25, 50%). N = 3 was chosen; however, the use
of the price $5.50 resulted in no onions being planted. The price was therefore
increased to $6.00, which matches the expected price assumed in the base case
model. The categories used in models that incorporate price uncertainty are ‘hi’ =
$12.25, ‘med’ = $6.00, and ‘lo’ = $2.50.
241
1.1
1
0.9
0.8
F(x)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
Onion Price ($/cwt) (2004$)
Figure B.3. Cumulative distribution function for the random variable
“onion price,” estimated with historical data from the study area.
14
242
Appendix C. Prevented planting model’s parameters and profit.
Table C.1. Approved yields used in the calculation of a prevented planting
payment.
Insured Crop Approved Yield
(yield Unit)
Per Acre
Onion (cwt)
550
Potato (cwt)
415
Sugar beet (T)
31
Wheat (bu)
130
Table C.2. Available coverage levels for the multi-peril crop insurance contract.
Insured Crop
Onion
Potato
Sugar beet
Wheat
Coverage
Levels (%)
50
65
75
75
50
65
75
85
55
65
75
85
Table C.3. Price election ($/unit of production) assumed for each insured crop.
Insured Crop Price Elected
Onion (cwt)
3.25
Potato (cwt)
3.34
Sugar beet (T)
39.00
Wheat (bu)
3.22
Table C.4. Prevented planting coverage level assumed for each insured crop.
Insured Crop PP Coverage (%)
Onion
45
Potato
25
Sugar beet
45
Wheat
60
243
Table C.5. Discounted and undiscounted profit (i.e. returns to land and
management) ($2004) by scenario for the “subsidized prevented planting” model’s
solution. Scenarios are sorted by discounted profit (in ascending order).
Yr1
FULL
FULL
FULL
FULL
FULL
FULL
DRY
DRY
DRY
FULL
FULL
FULL
FULL
FULL
DRY
FULL
DRY
FULL
FULL
FULL
FULL
FULL
FULL
DRY
FULL
DRY
DRY
FULL
DRY
DRY
DRY
DRY
FULL
DRY
FULL
FULL
DRY
Yr2
FULL
FULL
FULL
FULL
DRY
FULL
DRY
DRY
FULL
DRY
FULL
DRY
FULL
FULL
FULL
DRY
FULL
DRY
FULL
DRY
FULL
DRY
FULL
FULL
FULL
FULL
FULL
DRY
DRY
DRY
DRY
DRY
DRY
FULL
DRY
FULL
DRY
Yr3
FULL
FULL
FULL
FULL
FULL
FULL
FULL
FULL
FULL
FULL
DRY
FULL
FULL
FULL
FULL
FULL
FULL
FULL
DRY
FULL
FULL
DRY
DRY
DRY
DRY
FULL
DRY
FULL
DRY
FULL
FULL
DRY
DRY
FULL
DRY
DRY
FULL
Yr4
FULL
FULL
FULL
FULL
FULL
DRY
FULL
FULL
FULL
FULL
FULL
FULL
DRY
DRY
FULL
DRY
FULL
FULL
FULL
DRY
DRY
FULL
FULL
FULL
DRY
FULL
FULL
DRY
FULL
DRY
FULL
FULL
FULL
DRY
DRY
FULL
DRY
Yr5
DRY
FULL
DRY
FULL
FULL
FULL
FULL
FULL
FULL
DRY
FULL
FULL
DRY
FULL
FULL
FULL
DRY
DRY
FULL
FULL
DRY
FULL
DRY
FULL
FULL
DRY
FULL
DRY
FULL
FULL
DRY
FULL
FULL
FULL
FULL
DRY
FULL
Yr6
FULL
FULL
DRY
DRY
FULL
FULL
DRY
FULL
DRY
FULL
FULL
DRY
FULL
DRY
FULL
FULL
FULL
DRY
DRY
DRY
DRY
FULL
FULL
DRY
FULL
DRY
FULL
FULL
DRY
FULL
FULL
FULL
DRY
FULL
FULL
DRY
DRY
Discounted π
577,023
577,719
587,111
587,808
591,589
593,457
594,659
599,603
600,244
601,013
601,431
601,678
601,814
601,959
602,003
603,967
609,294
611,101
611,520
612,469
613,307
614,424
616,081
616,675
616,859
617,820
618,435
619,111
619,604
619,649
620,381
621,557
624,513
625,018
625,595
627,574
628,151
Undiscounted π
818,768
819,723
833,060
834,016
837,423
841,405
840,008
847,012
846,666
850,348
851,304
851,715
852,820
853,450
849,158
854,431
859,062
864,641
865,596
866,476
869,103
867,800
871,564
868,343
872,351
871,141
870,835
875,154
873,681
874,457
875,893
876,448
882,093
880,865
883,016
887,846
886,502
244
Table C.5 (Cont.)
Yr1
FULL
DRY
FULL
FULL
DRY
DRY
DRY
DRY
FULL
FULL
DRY
FULL
DRY
DRY
FULL
DRY
FULL
DRY
DRY
DRY
FULL
DRY
DRY
DRY
DRY
DRY
DRY
Yr2
FULL
DRY
DRY
FULL
FULL
FULL
DRY
DRY
DRY
DRY
FULL
FULL
FULL
DRY
DRY
DRY
DRY
DRY
FULL
FULL
DRY
DRY
FULL
FULL
DRY
FULL
DRY
Yr3
DRY
FULL
FULL
DRY
FULL
FULL
DRY
FULL
DRY
DRY
DRY
DRY
FULL
DRY
DRY
FULL
DRY
DRY
DRY
DRY
DRY
DRY
DRY
DRY
DRY
DRY
DRY
Yr4
DRY
FULL
DRY
DRY
DRY
DRY
DRY
DRY
FULL
DRY
FULL
DRY
DRY
DRY
DRY
DRY
FULL
FULL
FULL
DRY
DRY
FULL
DRY
DRY
DRY
DRY
DRY
Yr5
FULL
DRY
DRY
DRY
DRY
FULL
FULL
DRY
DRY
FULL
DRY
DRY
DRY
FULL
DRY
DRY
DRY
DRY
DRY
FULL
DRY
DRY
FULL
DRY
DRY
DRY
DRY
Yr6
DRY
DRY
DRY
FULL
FULL
DRY
FULL
FULL
FULL
DRY
FULL
DRY
DRY
DRY
FULL
DRY
DRY
FULL
DRY
FULL
DRY
DRY
DRY
FULL
FULL
DRY
DRY
Discounted π
628,352
628,883
630,447
632,737
633,375
633,520
634,088
634,943
636,389
636,931
638,649
644,230
644,868
645,424
645,814
646,278
647,882
649,461
649,985
652,880
657,150
660,955
664,374
665,583
666,034
677,076
677,370
Undiscounted π
888,634
887,938
891,214
894,129
892,280
892,910
893,306
895,393
898,094
899,076
899,085
910,412
908,563
909,366
910,748
911,452
914,377
914,816
915,144
918,021
926,808
931,099
934,304
935,444
937,123
951,727
953,182
245
Table C.6. Discounted and undiscounted profit (i.e. returns to land and
management) ($2004) by scenario for the “unsubsidized prevented planting”
model’s solution. Scenarios are sorted by discounted profit (in ascending order).
Yr1
DRY
FULL
DRY
DRY
DRY
DRY
FULL
FULL
DRY
DRY
FULL
DRY
DRY
FULL
DRY
FULL
FULL
FULL
FULL
FULL
FULL
DRY
DRY
FULL
DRY
DRY
FULL
DRY
FULL
FULL
DRY
Yr2
FULL
FULL
FULL
DRY
DRY
FULL
FULL
FULL
FULL
DRY
FULL
DRY
DRY
DRY
DRY
FULL
DRY
DRY
DRY
FULL
DRY
FULL
DRY
FULL
FULL
FULL
FULL
DRY
DRY
FULL
DRY
Yr3
FULL
FULL
FULL
FULL
FULL
FULL
FULL
FULL
FULL
FULL
FULL
FULL
DRY
FULL
DRY
FULL
FULL
FULL
FULL
FULL
FULL
DRY
FULL
FULL
DRY
FULL
DRY
FULL
FULL
FULL
FULL
Yr4
FULL
FULL
FULL
FULL
FULL
FULL
FULL
FULL
FULL
FULL
FULL
FULL
FULL
FULL
FULL
DRY
FULL
FULL
DRY
DRY
FULL
FULL
DRY
DRY
FULL
DRY
FULL
DRY
DRY
DRY
DRY
Yr5
FULL
DRY
FULL
FULL
FULL
DRY
DRY
FULL
DRY
DRY
FULL
DRY
FULL
FULL
FULL
FULL
FULL
DRY
FULL
DRY
DRY
FULL
FULL
FULL
FULL
FULL
FULL
FULL
FULL
DRY
DRY
Yr6
DRY
FULL
FULL
DRY
FULL
DRY
DRY
FULL
FULL
DRY
DRY
FULL
DRY
FULL
FULL
FULL
DRY
DRY
FULL
FULL
FULL
DRY
FULL
DRY
FULL
FULL
FULL
DRY
DRY
DRY
FULL
Discounted π
574,026
575,409
575,785
576,854
577,639
577,914
578,314
578,775
579,673
581,487
581,680
582,272
583,858
584,039
584,642
585,681
585,816
587,404
589,230
589,569
589,630
589,963
589,976
590,704
591,722
592,892
593,546
593,713
594,253
594,592
594,941
Undiscounted π
814,098
816,760
816,590
819,004
820,115
819,431
820,876
821,377
821,923
825,358
825,493
826,470
828,296
828,177
829,408
831,086
830,694
832,836
835,319
836,419
835,988
835,693
837,086
838,203
838,186
840,295
841,374
842,380
842,436
843,535
843,895
246
Table C.6 (Cont.)
Yr1
DRY
FULL
FULL
DRY
DRY
FULL
FULL
FULL
DRY
FULL
DRY
DRY
DRY
DRY
FULL
FULL
DRY
FULL
FULL
FULL
DRY
DRY
FULL
FULL
DRY
FULL
FULL
DRY
FULL
FULL
DRY
DRY
DRY
Yr2
FULL
DRY
DRY
FULL
DRY
DRY
FULL
FULL
FULL
FULL
FULL
DRY
DRY
DRY
DRY
DRY
FULL
DRY
DRY
FULL
DRY
DRY
DRY
FULL
FULL
FULL
DRY
DRY
FULL
DRY
FULL
FULL
FULL
Yr3
FULL
FULL
DRY
FULL
DRY
DRY
DRY
DRY
DRY
DRY
DRY
FULL
DRY
DRY
FULL
DRY
FULL
DRY
DRY
DRY
DRY
DRY
DRY
DRY
DRY
DRY
DRY
DRY
DRY
DRY
DRY
DRY
DRY
Yr4
DRY
DRY
FULL
DRY
DRY
FULL
FULL
FULL
FULL
FULL
FULL
DRY
DRY
FULL
DRY
DRY
DRY
FULL
DRY
DRY
DRY
FULL
FULL
DRY
DRY
DRY
DRY
DRY
DRY
DRY
DRY
DRY
DRY
Yr5
FULL
DRY
FULL
DRY
DRY
FULL
FULL
DRY
DRY
DRY
DRY
DRY
FULL
DRY
DRY
FULL
DRY
DRY
FULL
FULL
FULL
DRY
DRY
FULL
FULL
DRY
DRY
DRY
DRY
DRY
FULL
DRY
DRY
Yr6
DRY
FULL
FULL
FULL
FULL
DRY
DRY
FULL
FULL
DRY
DRY
DRY
FULL
FULL
DRY
FULL
DRY
FULL
DRY
FULL
DRY
DRY
DRY
DRY
FULL
FULL
FULL
DRY
DRY
DRY
DRY
FULL
DRY
Discounted π
595,343
595,613
595,758
595,795
596,147
596,248
596,452
596,616
596,859
598,092
598,335
598,678
598,940
599,841
600,636
600,690
600,818
601,383
603,140
603,735
603,963
604,864
605,420
606,185
606,327
606,443
607,783
608,093
608,894
610,233
611,351
614,406
616,857
Undiscounted π
843,767
844,074
843,981
844,231
844,989
844,675
845,490
845,799
845,586
847,890
847,677
849,190
849,132
850,492
851,190
850,811
851,348
851,630
854,283
855,622
856,249
857,609
857,350
859,094
858,414
859,337
860,540
861,913
862,809
864,012
865,531
869,540
873,011
247
Appendix D. Price uncertainty model’s profit.
Table D1. Discounted and undiscounted profit (i.e. returns to land and
management) ($2004) by scenario for the “price uncertainty” model’s solution.
Yr1
DRY
DRY
DRY
FULL
FULL
FULL
DRY
DRY
DRY
DRY
DRY
DRY
FULL
FULL
FULL
FULL
FULL
FULL
DRY
DRY
DRY
FULL
FULL
FULL
DRY
DRY
DRY
FULL
FULL
FULL
DRY
DRY
Yr1
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
Yr2
DRY
DRY
DRY
DRY
DRY
DRY
DRY
DRY
DRY
FULL
FULL
FULL
DRY
DRY
DRY
FULL
FULL
FULL
FULL
FULL
FULL
FULL
FULL
FULL
DRY
DRY
DRY
DRY
DRY
DRY
DRY
DRY
Yr2
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
MED
MED
MED
MED
MED
MED
MED
MED
Yr3
DRY
DRY
DRY
DRY
DRY
DRY
FULL
FULL
FULL
DRY
DRY
DRY
FULL
FULL
FULL
DRY
DRY
DRY
FULL
FULL
FULL
FULL
FULL
FULL
DRY
DRY
DRY
DRY
DRY
DRY
FULL
FULL
Yr3
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
Discounted π Undiscounted π
-755,400
-797,514
-755,400
-797,514
-755,400
-797,514
-743,135
-784,220
-743,135
-784,220
-743,135
-784,220
-726,754
-762,456
-726,754
-762,456
-726,754
-762,456
-708,212
-741,604
-708,212
-741,604
-708,212
-741,604
-700,439
-731,967
-700,439
-731,967
-700,439
-731,967
-694,109
-726,062
-694,109
-726,062
-694,109
-726,062
-679,566
-706,546
-679,566
-706,546
-679,566
-706,546
-651,414
-673,809
-651,414
-673,809
-651,414
-673,809
-539,224
-541,368
-539,224
-541,368
-539,224
-541,368
-527,306
-528,499
-527,306
-528,499
-527,306
-528,499
-509,271
-504,711
-509,271
-504,711
248
Table D1 (Cont.)
Yr1
DRY
DRY
DRY
DRY
FULL
FULL
FULL
FULL
FULL
FULL
DRY
DRY
DRY
FULL
FULL
FULL
DRY
DRY
DRY
FULL
FULL
FULL
DRY
DRY
DRY
DRY
DRY
DRY
FULL
FULL
FULL
FULL
FULL
FULL
Yr1
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
Yr2
DRY
FULL
FULL
FULL
DRY
DRY
DRY
FULL
FULL
FULL
FULL
FULL
FULL
FULL
FULL
FULL
DRY
DRY
DRY
DRY
DRY
DRY
DRY
DRY
DRY
FULL
FULL
FULL
DRY
DRY
DRY
FULL
FULL
FULL
Yr2
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
Yr3
FULL
DRY
DRY
DRY
FULL
FULL
FULL
DRY
DRY
DRY
FULL
FULL
FULL
FULL
FULL
FULL
DRY
DRY
DRY
DRY
DRY
DRY
FULL
FULL
FULL
DRY
DRY
DRY
FULL
FULL
FULL
DRY
DRY
DRY
Yr3
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
Discounted π Undiscounted π
-509,271
-504,711
-492,384
-485,884
-492,384
-485,884
-492,384
-485,884
-484,611
-476,247
-484,611
-476,247
-484,611
-476,247
-480,118
-472,589
-480,118
-472,589
-480,118
-472,589
-463,738
-450,825
-463,738
-450,825
-463,738
-450,825
-437,422
-420,336
-437,422
-420,336
-437,422
-420,336
-154,164
-85,150
-154,164
-85,150
-154,164
-85,150
-140,062
-69,607
-140,062
-69,607
-140,062
-69,607
-125,518
-50,091
-125,518
-50,091
-125,518
-50,091
-106,976
-29,239
-106,976
-29,239
-106,976
-29,239
-97,366
-17,354
-97,366
-17,354
-97,366
-17,354
-92,873
-13,697
-92,873
-13,697
-92,873
-13,697
249
Table D1 (Cont.)
Yr1
DRY
DRY
DRY
FULL
FULL
FULL
DRY
DRY
DRY
FULL
FULL
FULL
DRY
DRY
DRY
DRY
DRY
DRY
FULL
FULL
FULL
FULL
FULL
FULL
DRY
DRY
DRY
FULL
FULL
FULL
DRY
DRY
DRY
FULL
Yr1
LO
LO
LO
LO
LO
LO
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
Yr2
FULL
FULL
FULL
FULL
FULL
FULL
DRY
DRY
DRY
DRY
DRY
DRY
FULL
FULL
FULL
DRY
DRY
DRY
FULL
FULL
FULL
DRY
DRY
DRY
FULL
FULL
FULL
FULL
FULL
FULL
DRY
DRY
DRY
DRY
Yr2
HI
HI
HI
HI
HI
HI
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
MED
MED
MED
MED
Yr3
FULL
FULL
FULL
FULL
FULL
FULL
DRY
DRY
DRY
DRY
DRY
DRY
DRY
DRY
DRY
FULL
FULL
FULL
DRY
DRY
DRY
FULL
FULL
FULL
FULL
FULL
FULL
FULL
FULL
FULL
DRY
DRY
DRY
DRY
Yr3
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
Discounted π
-78,330
-78,330
-78,330
-50,178
-50,178
-50,178
204,266
204,266
204,266
227,071
227,071
227,071
229,836
229,836
229,836
237,197
237,197
237,197
258,731
258,731
258,731
260,002
260,002
260,002
272,531
272,531
272,531
291,662
291,662
291,662
420,094
420,094
420,094
438,008
Undiscounted π
5,819
5,819
5,819
38,556
38,556
38,556
305,192
305,192
305,192
331,393
331,393
331,393
335,251
335,251
335,251
345,494
345,494
345,494
368,905
368,905
368,905
371,695
371,695
371,695
387,503
387,503
387,503
409,207
409,207
409,207
560,913
560,913
560,913
581,128
250
Table D1 (Cont.)
Yr1
FULL
FULL
DRY
DRY
DRY
DRY
DRY
DRY
FULL
FULL
FULL
FULL
FULL
FULL
DRY
DRY
DRY
FULL
FULL
FULL
DRY
DRY
DRY
FULL
FULL
FULL
DRY
DRY
DRY
DRY
DRY
DRY
FULL
FULL
Yr1
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
Yr2
DRY
DRY
FULL
FULL
FULL
DRY
DRY
DRY
FULL
FULL
FULL
DRY
DRY
DRY
FULL
FULL
FULL
FULL
FULL
FULL
DRY
DRY
DRY
DRY
DRY
DRY
FULL
FULL
FULL
DRY
DRY
DRY
FULL
FULL
Yr2
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
Yr3
DRY
DRY
DRY
DRY
DRY
FULL
FULL
FULL
DRY
DRY
DRY
FULL
FULL
FULL
FULL
FULL
FULL
FULL
FULL
FULL
DRY
DRY
DRY
DRY
DRY
DRY
DRY
DRY
DRY
FULL
FULL
FULL
DRY
DRY
Yr3
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
Discounted π
438,008
438,008
448,894
448,894
448,894
453,025
453,025
453,025
474,560
474,560
474,560
479,504
479,504
479,504
484,686
484,686
484,686
507,491
507,491
507,491
797,574
797,574
797,574
828,307
828,307
828,307
830,434
830,434
830,434
840,270
840,270
840,270
852,040
852,040
Undiscounted π
581,128
581,128
594,924
594,924
594,924
601,215
601,215
601,215
624,626
624,626
624,626
631,912
631,912
631,912
638,727
638,727
638,727
664,928
664,928
664,928
1,007,855
1,007,855
1,007,855
1,043,758
1,043,758
1,043,758
1,046,835
1,046,835
1,046,835
1,060,107
1,060,107
1,060,107
1,071,568
1,071,568
251
Table D1 (Cont.)
Yr1
FULL
FULL
FULL
FULL
DRY
DRY
DRY
FULL
FULL
FULL
DRY
DRY
DRY
FULL
FULL
FULL
DRY
DRY
DRY
DRY
DRY
DRY
FULL
FULL
FULL
FULL
FULL
FULL
DRY
DRY
DRY
FULL
FULL
FULL
Yr1
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
Yr2
FULL
DRY
DRY
DRY
FULL
FULL
FULL
FULL
FULL
FULL
DRY
DRY
DRY
DRY
DRY
DRY
DRY
DRY
DRY
FULL
FULL
FULL
FULL
FULL
FULL
DRY
DRY
DRY
FULL
FULL
FULL
FULL
FULL
FULL
Yr2
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
LO
Yr3
DRY
FULL
FULL
FULL
FULL
FULL
FULL
FULL
FULL
FULL
DRY
DRY
DRY
DRY
DRY
DRY
FULL
FULL
FULL
DRY
DRY
DRY
DRY
DRY
DRY
FULL
FULL
FULL
FULL
FULL
FULL
FULL
FULL
FULL
Yr3
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
Discounted π
852,040
861,238
861,238
861,238
871,930
871,930
871,930
894,735
894,735
894,735
1,891,187
1,891,187
1,891,187
1,920,276
1,920,276
1,920,276
1,933,882
1,933,882
1,933,882
1,934,116
1,934,116
1,934,116
1,953,773
1,953,773
1,953,773
1,962,971
1,962,971
1,962,971
1,976,811
1,976,811
1,976,811
1,996,469
1,996,469
1,996,469
Undiscounted π
1,071,568
1,084,060
1,084,060
1,084,060
1,097,620
1,097,620
1,097,620
1,123,820
1,123,820
1,123,820
2,242,553
2,242,553
2,242,553
2,276,128
2,276,128
2,276,128
2,294,806
2,294,806
2,294,806
2,293,302
2,293,302
2,293,302
2,315,888
2,315,888
2,315,888
2,328,380
2,328,380
2,328,380
2,345,555
2,345,555
2,345,555
2,368,140
2,368,140
2,368,140
252
Table D1 (Cont.)
Yr1
DRY
DRY
DRY
FULL
FULL
FULL
DRY
DRY
DRY
DRY
DRY
DRY
FULL
FULL
FULL
FULL
FULL
FULL
DRY
DRY
DRY
FULL
FULL
FULL
DRY
DRY
DRY
FULL
FULL
FULL
DRY
DRY
DRY
Yr1
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
Yr2
DRY
DRY
DRY
DRY
DRY
DRY
DRY
DRY
DRY
FULL
FULL
FULL
FULL
FULL
FULL
DRY
DRY
DRY
FULL
FULL
FULL
FULL
FULL
FULL
DRY
DRY
DRY
DRY
DRY
DRY
DRY
DRY
DRY
Yr2
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
MED
HI
HI
HI
HI
HI
HI
HI
HI
HI
Yr3
DRY
DRY
DRY
DRY
DRY
DRY
FULL
FULL
FULL
DRY
DRY
DRY
DRY
DRY
DRY
FULL
FULL
FULL
FULL
FULL
FULL
FULL
FULL
FULL
DRY
DRY
DRY
DRY
DRY
DRY
FULL
FULL
FULL
Yr3
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
Discounted π Undiscounted π
2,107,015
2,498,274
2,107,015
2,498,274
2,107,015
2,498,274
2,136,104
2,531,848
2,136,104
2,531,848
2,136,104
2,531,848
2,149,711
2,550,526
2,149,711
2,550,526
2,149,711
2,550,526
2,149,944
2,549,023
2,149,944
2,549,023
2,149,944
2,549,023
2,169,601
2,571,609
2,169,601
2,571,609
2,169,601
2,571,609
2,178,800
2,584,101
2,178,800
2,584,101
2,178,800
2,584,101
2,192,640
2,601,275
2,192,640
2,601,275
2,192,640
2,601,275
2,212,297
2,623,861
2,212,297
2,623,861
2,212,297
2,623,861
2,490,586
2,952,670
2,490,586
2,952,670
2,490,586
2,952,670
2,529,439
2,998,195
2,529,439
2,998,195
2,529,439
2,998,195
2,533,282
3,004,922
2,533,282
3,004,922
2,533,282
3,004,922
253
Table D1 (Cont.)
Yr1
DRY
DRY
DRY
FULL
FULL
FULL
FULL
FULL
FULL
DRY
DRY
DRY
FULL
FULL
FULL
Yr1
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
Yr2
FULL
FULL
FULL
FULL
FULL
FULL
DRY
DRY
DRY
FULL
FULL
FULL
FULL
FULL
FULL
Yr2
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
HI
Yr3
DRY
DRY
DRY
DRY
DRY
DRY
FULL
FULL
FULL
FULL
FULL
FULL
FULL
FULL
FULL
Yr3
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
HI
MED
LO
Discounted π
2,544,996
2,544,996
2,544,996
2,555,009
2,555,009
2,555,009
2,562,370
2,562,370
2,562,370
2,573,642
2,573,642
2,573,642
2,597,705
2,597,705
2,597,705
Undiscounted π
3,017,470
3,017,470
3,017,470
3,028,253
3,028,253
3,028,253
3,038,497
3,038,497
3,038,497
3,052,528
3,052,528
3,052,528
3,080,505
3,080,505
3,080,505
254
Appendix E. List of electronic files provided.
Computer Program Files (GAMS):
Base case model (binary): ip_base.gms
Base case model (continuous): cont_base.gms
Model Solution Files (Excel)
Base case solution (binary): ip_base.xcl
Base case solution (continuous): cont_base.xcl
255